Mr Bashir wrote to the men, requesting a meeting and is understood to be seeing them later today when a deal is expected to be agreed.
The interview is likely to be broadcast as part of a Panorama programme when the public inquiry into Stephen's death finishes later this year.
The five men, David Norris, 21, Neil Acourt, 22, his brother Jamie, 21, Luke Knight, 20, and Gary Dobson, 22, have so far refused to answer direct questions about the stabbing at a south London bus stop in April 1993. This June, under duress, they appeared before the public inquiry, but gave only evasive answers when cross-examined.
If they did agree to be interviewed on television, the men would presumably be asked the one question that the inquiry was prevented by a High Court injunction from posing: did they kill Stephen?
Mr Bashir was yesterday unavailable for comment, but in his letter to Jamie Acourt, on BBC-headed paper, he asks for a "brief meeting".
He writes: "For what it's worth I am the BBC reporter who recently interviewed Miss Louise Woodward ... I also conducted an interview with the late Diana, Princess of Wales. Both were transmitted within the BBC's flagship current affairs programme Panorama.
"I was wondering if there was any chance of a brief meeting with you? I confirm that any conversation would be strictly confidential and would not be published, transmitted or repeated to a third party without your prior consent."
A spokesman for the Lawrence inquiry team said yesterday "I can't believe that [the five men] will answer questions any more fully than they did here, but certainly we will be following this development with a great deal of interest."
The Stephen Lawrence Campaign, set up to uncover the truth about his killing, last night condemned any plans to interview the five suspects, saying that Mr Bashir could hope to achieve nothing useful.
Ros Howell, chairman of the Lewisham Race Equality Group and a member of the campaign, said: "I would like to ask Mr Bashir what he thinks he would be doing by giving these men air-time. They had a fine opportunity to say whatever it is they are keeping back or not keeping back at the inquiry this summer and yet they said nothing of use ... I would find it very distasteful for them to appear on screen to make publicity from Stephen's death."
Neil Acourt, Gary Dobson and Luke Knight were cleared of murdering Stephen and cannot be retried. Charges against Jamie Acourt and David Norris were dropped.
Their apparent agreement to be questioned by Mr Bashir also raises doubts about the terms of the interview. Although he has been granted two of the most sought-after interviews in recent years, he has been accused of not being tough enough on his subjects.
Max Clifford, the public relations consultant, is thought to be behind the five men's decision to talk publicly. He said yesterday that they had consulted him and although he declined to represent them, he suggested Mr Bashir.
Peter Horrocks, the editor of Panorama, confirmed that a documentary team was researching a programme about Stephen Lawrence but refused to discuss details.