He is due to take over next month, but concerns have already been raised about his appointment in the wake of the Jimmy Savile scandal.
Just last week the newspaper's public editor Margaret Sullivan, whose role is to oversee standards, questioned Mr Thompson's claim that he had not been formally told of a Newsnight investigation into Savile's activities, which was eventually shelved.
NYT columnist Joe Nocera said in the paper today that Mr Thompson appeared "wilfully ignorant" of the issues and questioned what sort of an organisation the BBC was under his command. He added there was "plainly" a cover-up.
There have been repeated suggestions, which the corporation denies, that the BBC pulled the Newsnight probe because it had already lined up tribute programmes to the late TV and radio presenter.
An inquiry is being undertaken, led by former Sky News chief Nick Pollard, into the decision-making process over the dropped investigation. Police are also investigating Savile's activities and he is now believed to have been of the UK's most prolific abusers, with around 300 possible victims.
Questioning his role, Nocera wrote in the New York Times: "Thompson now says that at a cocktail party last December, a BBC reporter said to him, 'You must be worried about the Newsnight investigation into Jimmy Savile.
"Soon thereafter, Thompson asked his underlings about the investigation and was told that it had been killed - for journalistic reasons. He claims to have inquired no further, not even to ask what the investigation was about."
Mr Thompson stepped down from his post as director-general last month and his successor George Entwistle was engulfed in the scandal just days into the job.
Sullivan said of Mr Thompson last week: "It's worth considering now whether he is the right person for the job, given this turn of events."
A further inquiry is being carried out into the "culture and practices" of the BBC during the years Savile worked there, led by Dame Janet Smith.
Deputy Prime Minister Nick Clegg has said "heads will need to roll" at the BBC if it is discovered that abuse was ignored.
"Serious questions need to be asked and if after we find out what's happened, it's clear that people have turned a blind eye or, worse still, connived with it then of course they're going to have to be held to account and - if that turns out to be the case - heads will need to roll of course," he told ITV's The Agenda.
Yesterday it emerged that Savile was barred from any involvement with the BBC's Children In Need charity.
Sir Roger Jones, a former chairman of the charity, said he had been uncomfortable about allowing Savile to have any association with their work.
Although he had "no evidence" that Savile was up to anything, he said he behaved strangely, adding: "I think we all recognised he was a pretty creepy sort of character."
And Britain's most senior police officer said if allegations about Jimmy Savile had been linked while he was alive, they would have exposed "a pattern of behaviour".
Metropolitan Police Commissioner Bernard Hogan-Howe said police and other organisations had not connected a number of separate claims made about Savile's allegedly predatory actions.