Mr. Giuliani would not comment on the contents but confirmed that the president’s lawyers had not rejected an interview altogether and had instead tried for an “avenue” that could work.

The response indicated how far apart the two sides remain. Mr. Mueller wants to question the president on a range of issues, chiefly his associates’ contacts with Russia; possible coordination between his campaign and Moscow’s election interference; and the intent behind presidential actions that could be construed as attempts to obstruct the inquiry, including firing the F.B.I. director.

But the president’s lawyers are concerned that if he is interviewed, Mr. Trump could perjure himself. That concern is in part driving the continuing negotiations. They had been prepared last week to tell Mr. Mueller that Mr. Trump would decline an interview, but the president, who believes he can convince Mr. Mueller that he is innocent, pushed his lawyers to continue negotiating.

By making another counterproposal after months of promises that they were only weeks away from deciding about an interview, Mr. Trump’s lawyers run the risk that Mr. Mueller could conclude that they are negotiating in bad faith to prolong the investigation. In a meeting with Mr. Trump’s lawyers this year, Mr. Mueller threatened to take the extraordinary step of subpoenaing the president to testify before a grand jury if he did not sit for a voluntary interview.

Mr. Trump’s lawyers indicated that the negotiations were winding down, setting the stage for a possible subpoena. The talks were nearing an impasse, Mr. Giuliani said on Wednesday when he called into a radio show hosted by Mr. Sekulow.