A spokesman for Mueller declined to comment.
"An investigation could be done expeditiously," the defense attorneys argue, given that congressional committees and the Justice Department are looking into leaks of information related to the Russia inquiry.
Manafort's lawyers do not specifically accuse the Special Counsel's office of being the source of any leaks.
In fact, they argue that the leaks that came before Mueller's appointment last May were part of a "false narrative" designed in part to "garner support" for the creation of a special counsel investigation. But they say the disclosures also were "clearly intended to unfairly prejudice the jury pool" by repeatedly linking Manafort to Russia.
The lawyers cite seven media reports referencing the investigation of Manafort's ties to a Russian-backed Ukrainian politician between October 2016 and August 2017.
The first indictment against Manafort, accusing him of conspiracy to launder money, making false statements and other charges in connection with his work in Ukraine, was unsealed in October 2017.
Manafort's lawyers list the New York Times, CNN, NBC, AP, BuzzFeed and the L.A. Times as media outlets they claim published leaked information. The motion says those outlets are a "small sampling" of those it contends have received such information.
In particular, Manafort's lawyers note a New York Times report from February 2017 that a phone call was intercepted between Manafort and someone in the Russian Government or intelligence world. The story was sourced to "four current and former American officials."
Manafort's lawyers said they have asked for any material showing surveillance or interception of conversations between their client and Russian government or intelligence officials and have been told that none exists.
During testimony before a congressional committee last year, former FBI Director James Comey said the New York Times article was inaccurate. "In the main, it was not true," he said.
The newspaper has previously stood by the report, with Executive Editor Dean Baquet saying in a statement that the paper "had numerous sources confirming this story."