Thompson, named in August as the US media firm's president and chief executive, remains part of the news even though he has denied any role in shelving an investigative report into sex abuse by the late BBC star Jimmy Savile.
The Times has reaffirmed support for Thompson and he has said the scandals at the BBC would not affect his new job.
But as Thompson was readying his entry in New York, his successor as BBC director-general, George Entwistle, resigned on Saturday after the broadcaster's flagship news program wrongly implicated a British politician in a separate child sex abuse scandal.
"Like many people, I'm very saddened by recent events at the BBC but I believe the BBC is the world's greatest broadcaster and I've got no doubt that it will once again regain the public's trust both in the UK and around the world," Thompson told ITV, another British broadcaster, as he entered The Times headquarters.
He added that the scandals and probes at the BBC "will not in any way affect my job" at The Times.
Times chairman Arthur Sulzberger Jr last month reaffirmed support for Thompson, saying the ex-BBC chief provided a "detailed account" of his knowledge in the Savile case which showed "he played no role" in the cancellation of the investigative report.
But that has not stopped rumblings inside and outside the prestigious US daily about whether Thompson is the right leader for a news organization facing its own struggles.
Times business columnist Joe Nocera openly questioned whether Thompson was "the right man for the job" in New York.
"Since early October, all anybody has asked about Thompson are those two most damning of questions: what did he know, and when did he know it?" Nocera wrote in an October 29 column.
"For the sake of Times employees -- not to mention the readers who want to see a vibrant New York Times Company -- let's hope his faith in Thompson is warranted. Otherwise, the BBC won't be the only organization being asked tough questions about its judgment."
The Times meanwhile has been placed in an awkward position of writing about a BBC probe and its own top executive's role in the scandal.
Times public editor Margaret Sullivan said the daily has an obligation to "aggressively cover" the story even if Thompson is part of the news.
"His integrity and decision-making are bound to affect The Times and its journalism -- profoundly," Sullivan said in a recent column.
"I hope The Times rises to the challenge and thoroughly reports what it finds."
Others have been even more blunt about what The Times should do.
Ken Doctor, an analyst with the media research firm Outsell, said Thompson "should step aside" for the sake of the newspaper.
"The global value of the Times's brand and its trustworthiness must trump any one person's job or future," Doctor said.
Doctor said that "there are no smoking guns to show willful cover-up," but that "there is ample evidence that Thompson missed a number of opportunities to right his institution."
But Emily Bell, director of the Tow Center for Digital Journalism at Columbia University, said the criticism of Thompson is "odd."
"Thompson was not hired to oversee editorial at the NYT, in fact expressly not," Bell said in a tweet.
The Times, which is America's most prestigious newspaper as well as a growing online news power, has said it had recruited Thompson for his ability to develop non-traditional news products.
Glenn Greenwald, a columnist and blogger for the British daily The Guardian, said in a tweet that The Times "deserves much credit for leading the way in investigating its own incoming CEO."