The move represents a major development after months of negotiations and signals that the Mueller investigation could be entering a final phase with regard to the President, the sources told CNN on Thursday.
Trump's lawyers are preparing written responses, in part relying on documents previously provided to Mueller.
"We are in continuing discussions with the Special Counsel and we do not comment on those discussions," said Trump's attorney Jay Sekulow.
Asked on Thursday about answering Mueller's questions, Trump again signalled his willingness to sit down for an interview with the Special Counsel or provide written responses -- the option much preferred by his attorneys.
"It seems ridiculous that I'd have to do it when everybody says there's no collusion, but I'll do what is necessary to get it over with," Trump told Fox News.
(With inputs from agencies.)