As they ate, the president and Mr. Comey made small talk about the election and the crowd sizes at Mr. Trump’s rallies. The president then turned the conversation to whether Mr. Comey would pledge his loyalty to him.

Mr. Comey declined to make that pledge. Instead, Mr. Comey has recounted to others, he told Mr. Trump that he would always be honest with him, but that he was not “reliable” in the conventional political sense.

The White House says this account is not correct. And Mr. Trump, in an interview on Thursday with NBC, described a far different dinner conversation with Mr. Comey in which the director asked to have the meeting and the question of loyalty never came up. It was not clear whether he was talking about the same meal, but they are believed to have had only one dinner together.

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By Mr. Comey’s account, his answer to Mr. Trump’s initial question apparently did not satisfy the president, the associates said. Later in the dinner, Mr. Trump again said to Mr. Comey that he needed his loyalty.

Mr. Comey again replied that he would give him “honesty” and did not pledge his loyalty, according to the account of the conversation.