Mueller's office agreed to receive written answers to questions about whether or not the Trump campaign worked with Russia in its efforts to interfere in the 2016 election.
The source told NBC that discussions are "moving in a positive direction."
This is the latest development in months-long negotiations between the two parties about a potential interview with the president.
The Times reported that the letter, however, did not seek written responses to questions about whether Trump obstructed justice. It also did not say that Mueller was giving up on his pursuit of an interview with the president, the Times reported.
The special counsel and Trump's legal team have been negotiating a potential interview with the president for months.
Trump's lawyer Jay Sekulow said in a statement to NBC, "We have an ongoing dialogue with the office of special counsel and we do not discuss the contents of those discussions."