Singh,83, at present the Deputy Leader of the Congress in the upper house, was believed to have been made the offer by Congress President Sonia Gandhi after she was authorised by the Parliamentary Party last week to choose leaders for both the houses.
The fact that Singh was made the offer indicates that Manmmohan Singh, who was the leader of the house when he was prime minister for the last 10 years, may be disinclined to be the Leader of the Opposition, a post he had held during the Vajpayee government years.
Karan Singh, a scholar, is understood to have conveyed to Gandhi his inability to take up the post in view of his other commitments in the field of culture.
Other senior leaders in Rajya Sabha include A K Antony, Ghulam Nabi Azad, Janardhan Dwivedi, Digvijay Singh and Jairam Ramesh.
It is also not clear as to what the Congress President would do on the post of Leader of the Opposition in Lok Sabha, where it has got only 44 members after the recent elections.
The names doing the rounds for the post includes Kamal Nath, who has been elected for the ninth term this year, and Veerappa Moily, a former union minister and a chief minister.
Party insiders, however, say that a majority would like the post to be occupied by either Rahul or Sonia Gandhi.