Former New York City Mayor Rudy Giuliani introduces Republican presidential candidate Donald Trump at a rally on August 18, 2016 at the Charlotte Convention Center in Charlotte, North Carolina.

Special counsel Robert Mueller's office plans to complete its probe into alleged obstruction of the Russia inquiry by President Donald Trump by Sept. 1, the president's personal lawyer told The New York Times in an interview published Sunday.

While Mueller's ongoing broader probe is expected to continue, it suggests investigators may seek to avoid influencing public opinion ahead of the 2018 midterm elections.

Rudolph Giuliani told The Times that the special counsel's office shared its timeline about two weeks ago, in the middle of discussions about whether Trump would submit to questioning by Mueller.

"You don't want another repeat of the 2016 election where you get contrary reports at the end and you don't know how it affected the election," Giuliani told the publication, who shared similar remarks with NBC News.

Meanwhile, Reuters cited a source familiar with the probe who called the Sept. 1 deadline "entirely made-up" and "another apparent effort to pressure the special counsel to hasten the end of his work."

"He'll wrap it up when he thinks he's turned over every rock, and when that is will depend on how cooperative witnesses, persons of interest and maybe even some targets are, if any of those emerge, and on what new evidence he finds, not on some arbitrary, first-of-the-month deadline one of the president's attorneys cooks up," said the source, a U.S. official who spoke on condition of anonymity, according to Reuters.