President Trump pushed his lawyers in recent days to try once again to reach an agreement with the special counsel’s office about him sitting for an interview, flouting their advice that he should not answer investigators’ questions, three people briefed on the matter said on Wednesday.

Mr. Trump has told advisers he is eager to meet with investigators to clear himself of wrongdoing, the people said. In effect, he believes he can convince the investigators for the special counsel, Robert S. Mueller III, of his belief that their own inquiry is a “witch hunt.”

Mr. Mueller, whose team has negotiated the parameters of an interview with Mr. Trump’s lawyers for eight months, sent his latest proposal in a letter on Tuesday night, the three people said. Investigators stood firm on the scope of and topics for their questions for Mr. Trump: possible coordination between his associates and Russia’s election interference and whether he tried to obstruct the investigation.