Thompson won decathlon gold for Britain in 1980 and 1984 and is the choice of LOCOG, the organising committee chaired by Lord Coe, according to a report in the Daily Mail.
Coe is close friends with Thompson but has also said in the past that he considers the former multi-sports star to be "the greatest Olympian we have delivered.'
However, the paper adds that the British Olympic Association want the honour to go to Steve Redgrave who won five rowing golds for Britain.
The paper explains: "While Coe has no direct role in the selection process, his No 2 at LOCOG, Paul Deighton, is one of the two key players making the decision. The BOA chief executive and Team GB's chef de mission Andy Hunt is the other.
"Both men can hardly fail to be influenced by the views of their respective chairmen, Coe and the BOA's Redgrave-supporting Lord Moynihan."
The decision on who will light the flame will be kept a secret until the moment itself happens on July 27.
One previous contender who is unlikely to light the flame though is former football captain David Beckham who said the honour should go to an Olympian.
"I've always said, lighting the torch in the stadium is something that should be done by an Olympian... who has done incredible things for our country and won gold medals," Beckham said.
Beckham admitted that he was disappointed he would not be competing at the Games as a member of the Team GB football team after playing a big ambassador's role in helping London get the Games.
"Obviously, all the talk of me possibly performing in the Olympics, it would have been a very proud moment for me," said Beckham.
"Everyone knows how proud I am of representing my country and to do it in my home town on such a big stage would have been incredible so, of course, I'm disappointed, but life goes on.
"My family are healthy, I'm pretty healthy, so at the end of the day, I'll be there to support the GB team.
"It's going to be a proud moment to be there and know that I was part of bringing the Olympics to the East End of London."