The suggestion may seem arcane or fey - both Gordon Brown and William Hague have recently ridiculed the notion of "post-colonial guilt" and said Britain should get over it. But the subject is taxing David Cameron as he arrives in Mumbai this morning with one of the biggest trade delegations in British history, hoping to carve out a share of the world's second-fastest-growing economy for firms at home.
He believes our future prosperity may depend in part on how Britons and Indians get on with each other - "people-to-people contacts" in officialese - and that we will not achieve the "stronger, wider, deeper" relationship we need unless we stop assuming our "shared history" is enough.
The problem is that Britons and Indians see the "shared history" differently. To this country, India is the world's largest democracy, which we left behind on Independence Day in 1947; because of our historic relationship, India shares with us an independent judiciary, a free press, the English language, and our love of cricket. There's a legacy of colonial architecture and Merchant Ivory scenes of sahibs and memsahibs of the Raj clinking sundowners on their bungalow verandahs.
For many, the Anglo-Indian relationship is summed up in icons such as chicken tikka masala, now regarded by some as our national dish, a pint of Kingfisher, The Kumars at Number 42. And our diplomats take comfort in the fact that more than one million people in Britain are of Indian descent.
But for many Indians, the history they "share" with us is one of humiliation: bloody massacres, mass arrests, the suppression of democratic political movements and the supplanting of its indigenous cultures to create a servile, anglicised elite.
This history - passed down by grandparents and repeated in school textbooks - lives on in India's national memory. It surfaces in moments of tension, as happened last year when angry British MPs accused India of "ingratitude" over our aid donations when the government in New Delhi gave a pounds 13 billion fighter plane order to our French rivals.
And it's apparent in recent reports in Indian newspapers about Britain "clamping down" on the numbers of students allowed to come here or on visa restrictions for Indian workers.
The question is, how do we move beyond this latent resentment at a history we cannot change, and start making better memories? With this in mind, the Prime Minister is said to be considering voicing Britain's regret for the worst excesses of its empire rule during his three-day visit, for outrages like the 1919 Amritsar massacre, when up to a thousand peacefully protesting Indians, including women and children, were shot dead by British troops.
An unambiguous statement of regret may come as a relief to Sir James Bevan, the UK's high commissioner in India, who endured an excruciating moment shortly before Christmas following a speech to promote our common values as free-market democracies. An elderly Sikh gentleman stood and said he'd been jailed for campaigning for democracy and independence. Why hasn't Britain apologised for that?
The high commissioner squirmed. He said he hadn't been born at the time and that in any case he wanted to focus on the future. Sir James had no mandate to say otherwise, but the Sikh gentleman's heartfelt plea raises fundamental questions about what kind of country we are, and how we explain that the Britain which today sends its troops abroad to promote democracy once jailed Indians who politely demanded it. Why can't we look that man in the eye and tell him we're sorry for locking him up?
The common answer is that it would be a meaningless gesture. David Cameron wasn't personally responsible for Amritsar. Should he also say sorry to the Palestinians for Richard the Lionheart's Crusade? Ought the Italians to make reparations for the Roman conquest of Britannia? It would be absurd.
India, however, does fall into a different category. There are more than eight million people alive today who were at least 15 years old at the time of independence - and for many, the cruelties of the British Raj are not ancient history but living memory. They may be grateful for the railways we left behind or the parliamentary system we established, yet the memories of some are shaming.
One such person is Subadhra Khosla, an 85-year-old retired social worker who joined Gandhi's non-violent movement for independence as a child and was sentenced to three years "rigorous imprisonment" in Lahore when she was just 13.
Her "crime" was to join a sit-down protest against British rule during heavy monsoon rains on August 26 1942. "They took me, my mother, brother and younger sister to the police station and then the magistrates' court. I was given one year for each of three offences. I had to work hard in jail for one year and two months," she told me last week.
Far worse acts were committed by Britain in the name of empire. They may have largely been forgotten here, but in India the memories are still painfully raw. They include the Bengal famine during the Second World War, in which more than a million Indians were allowed to starve to death after their rice paddies were turned over to produce jute for sandbags. Sir Winston Churchill ignored pleas to divert food ships.
The historian and author William Dalrymple believes the truth of colonial rule around the world needs to be taught as part of the new British history curriculum. His children studied the Tudors and Germany under the Nazis "over and over again", but had not learnt of the atrocities carried out by Britain in India and Afghanistan. "Millions of people were killed, it [colonial rule] rested on a mountain of skulls, and people need to know that," he says.
Dalrymple had arrived in India in 1984 after watching The Jewel in the Crown, the ITV series about the last days of the Raj, "with the belief that they loved the British. Nothing in the world prepared me for the negative side of British rule.
"We have to say, 'Personally, I don't like what happened, I'm very sorry about what happened,' but we can't take responsibility for 600 years of history," he says.
Pavan Varma, author of Being Indian, believes that the Raj still survives in the minds and cultural habits of Indians today, and that one of the country's greatest challenges is to reconnect with the indigenous languages and cultures that were displaced by a policy of anglicisation.
"If the Prime Minister says, 'If there is anything in the past for any living Indian, we apologise,' we have no problem," he says.
Varma believes that India's main challenge, though, is to reclaim its cultural identity: "The Union flag comes down, the Tricolour goes up but when a country rules for a hundred years so much of that past sails into the future."
How Britain reconciles its colonial rule as it seeks a thriving trading and cultural relationship with India was recently addressed in a British Council-led initiative called "Re-imagine: India-UK cultural relations in the 21st century". The research project heard that while the two countries have 200 years of "shared history", they had become estranged and that the young in Britain and India today know little about contemporary life in the other.
Rob Lynes, head of the British Council in India, says the legacy of empire is less of an issue than the challenge of connecting with the next generation.
One of his contributors had told him that the UK needs to fall in love with India again. "There is a sense here that the UK has been complacent in its relationship with India and in some cases may have taken it for granted because of our shared history," he says.
We in this country should remember how much of that "shared history" played a crucial part in making us who we are. The United Kingdom came about in 1707 in no small part because members of Scotland's nobility wanted East India Company trading positions for their sons - the lure of India cemented the Act of Union.
India's spices and pickles changed our tastebuds, its words crept into our language, and its men fought our wars and safeguarded our independence.
Around two and a half million Indians fought alongside British forces in the Second World War, helping to defeat Rommel's forces in North Africa and halting the advance of Japanese troops at Kohima in north-east India in heroic hand-to-hand combat. By the end of the war Britain owed 1.25 billion pounds of its total 3 billion pounds war debt to India, but also much more that could not be quantified. Thirty Indian soldiers were awarded Victoria Crosses for their part in defending the freedoms we cherish today.
An apology for the worst excesses of empire will be appreciated, and for many it may be necessary, but it will not do as much for our future relationship as a genuine "thank you". Perhaps by stressing our gratitude for how India has shaped modern Britain, David Cameron could start to rekindle and put right what has often been a one-sided love.
Veteran Gandhian Subadhra Khosla says she doesn't need an apology for being jailed at 13. The English were "crushing the Indians" during the Raj, but "there's no need to say sorry", she says. "We should love each other. We are human beings, children of God and we should create goodwill."