Thompson is understood to have damaged his hopes of lighting the Olympic cauldron after he was accused of racism.
The Olympic champion, 53, was on BBC1’s The One Show when he made the remark about a torchbearer with a tattoo that read “Oylmpic”.
He joked the tattooist “must have been Irish” — before being told it was done in the US.
The BBC issued an apology after admitting dozens of complaints were made, adding Thompson’s comments “were clearly meant as a joke”.
The corporation wrote: “Thompson’s comments about this were clearly meant as a joke, but we apologise if any offence was caused; it certainly wasn’t our intention.
“Matt [Baker] corrected Daley straight away and Alex offered an on-air apology at the end of the show.”
Meanwhile, Lord Coe yesterday insisted his close friendship with Thompson would not affect the decision on who lights the cauldron on Friday.
Steve Redgrave, Kelly Holmes, and Roger Bannister are the favourites to light the Olympic cauldron at the opening ceremony.
One blogger wrote: “His hopes of lighting the flame may have just disappeared with that remark!”
The misspelt tattoo was in fact Jerri Peterson’s, a US-born woman chosen to carry the torch as part of this year’s relay.
Daley’s comment follows the arrests of 25 far-right protesters for anti-Irish comments at a march in Liverpool celebrating the life of James Larkin.