The special counsel had signaled a willingness to accept written answers on matters of collusion but Trump's attorney, Rudy Giuliani, has said repeatedly that president would not answer Mueller's questions on possible obstruction of justice.
The source would not explicitly say whether the President answered those questions other than to say that there are "responses" to all of the questions that were asked.
The president did not say Friday when he would turn his answers over to Mueller.
The White House fears that Mueller will indict some of the president's allies and family members, Politico reported, citing unnamed officials working for or in contact with the White House.
"I'm sure they're tricked up", Trump said.
The report comes as Mueller reportedly prepares to issue more indictments, which could come as soon as Friday.
Meanwhile, after being uncharacteristically subdued leading up to the midterm elections earlier this month, Trump used Twitter on Thursday to lay into Mueller, accusing the special counsel's team of "screaming and shouting at people" and "horribly threatening them to come up with the answers they want".
There should have never been a so-called investigation. "They have found no collusion and have gone absolutely nuts ..."
Tensions have risen even higher over the probe into Russian interference in the 2016 USA elections since Trump called for the resignation of attorney general Jeff Sessions and replaced him with Matthew Whitaker on an acting basis.
Mueller already has brought charges against a series of former Trump aides.
In the US, "collusion" is not a federal crime, although the Mueller probe has been looking into whether members of Trump's team, or the president himself, were involved in the obstruction of justice or conspiracy. It was reminiscent of his admission to NBC's Lester Holt past year that he ousted James Comey as FBI director to kill the investigation.
Mueller's office is investigating whether anyone in Trump's campaign conspired with the Russians who attacked the 2016 presidential election.