"It is time to bring this inquiry to a conclusion", Giuliani said.
His lawyer, Jay Sekulow, said the questions answered in writing were exclusively related to the issue of Russian Federation and did not take into account suggestions that the president has tried to obstruct the investigation - notably through his sacking of former FBI director James Comey.
"We answered every question they asked that was legitimately pre-election and focused on Russia", Giuliani said in an interview.
A spokesman for Mueller, who was appointed special counsel in May 2017, declined to comment.
Trump, who is also reportedly being investigated by the special counsel for possible obstruction of justice, confirmed last week that he would personally draft responses to investigators' questions.More news: Braun Strowman ties WWE record at Survivor Series
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Another of Trump's lawyers, Rudy Giuliani, said the lawyers continue to believe that "much of what has been asked raised serious constitutional issues and was beyond the scope of a legitimate inquiry". Since then, he had indicted or won guilty pleas from four former Trump campaign advisers and charged 12 Russian military officers with hacking Democratic emails that were later released and damaging Democratic challenger Hillary Clinton. There have been reports, however, that the president's legal team has been staunchly opposed to a personal encounter between Trump and Mueller and eventually talked the president out of it.
Trump's lawyers and Mueller have been in negotiations for months over whether Trump would be interviewed by Mueller in person, submit written responses to Mueller's questions or neither.
The president told reporters last week that he had prepared the responses himself. In response to Trump's attack, Schiff tweeted, "Wow, Mr. President, that's a good one".
In an interview that aired on Sunday, he told "Fox News Sunday" that he was unlikely to agree to an interview, explaining that "we've wasted enough time on this witch hunt and the answer is, probably, we're finished". But Justice Department leaders, including acting Attorney General Matthew Whitaker - who now oversees the investigation and has spoken pejoratively of it in the past - would have to sign off on such a move, and it's far from clear that they would.
It is unclear whether Mueller would subpoena him to testify, likely setting off a legal and political battle. "As other presidents have found out, they are not above the law".