"Although the precise timing is subject to change, the preparations are the clearest indication so far that the special counsel is ready to wrap up his nearly two-year probe", adds the article.
"That'll be totally up to the new attorney general", Mr. Trump said at the White House.
Trump has denied any collusion and repeatedly slammed the Russian Federation probe as a 'witch hunt.' Moscow has also denied any meddling.
Democrats have been concerned that Mr Barr, who has discretion over what is ultimately made public, will choose to limit disclosure, although the new Attorney-General has said he would seek to make public as much of Mr Mueller's findings as he could.
Barr previewed how he would handle the release of the Mueller report during his congressional confirmation hearing, saying, "The special counsel will prepare a summary report on any prosecutive or declination decisions and that that shall be confidential and shall be treated as any other declination or prosecutive material within the department". NBC News had previously said that the investigation was likely to wrap up in the middle of February.
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The brothers are set to divide their staff, after they had a number of meetings to discuss the new arrangements in recent weeks. The newspaper said the division will happen before the birth of Harry and wife Meghan Markle's first child this spring.
Although the sources say the investigation is coming to a close, plans could still change.
Rosenstein had always been expected to vacate his post after Trump's new attorney general, William Barr, took office.
Regulations call for Mueller to submit to the attorney general a confidential explanation as to why he chose to charge certain individuals, as well as who else he investigated and why he decided not to charge those people.
The end of the special counsel's probe would not mean the end of criminal investigations connected to the president.
And the grand jury that Mueller's prosecutors used to return indictments of longtime Trump confidant Roger Stone, former Trump campaign chairman Paul Manafort, and several Russians hasn't apparently convened since January 24 the day it approved the criminal charges against Stone. The network cited a senior DOJ official in its report.
"His years of outstanding legal and management experience make him an excellent choice to succeed Deputy Attorney General Rod Rosenstein, who has served the Department of Justice over many years with dedication and distinction", Barr added.