Aides have counseled the president to stay off Twitter and focus on other aspects of his job.
In a third tweet, posted from the "very rustic" Camp David where the president is spending Father's Day weekend, Trump touted a poll released Friday showing his approval rating reached 50% for the first time in Rasmussen surveys.
Hours later, a source close to Mr Trump's outside legal team said Mr Trump did not intend his tweet to be confirmation of the investigation but rather was reacting to a Washington Post story on Wednesday about the probe.
He continued: "Americans should be skeptical about anonymous allegations".
Mueller spokesman Peter Carr tells NPR's Carrie Johnson, "We'll decline to comment".
The transition official said the organization has also separately asked the General Services Administration to preserve records from the Trump transition that were transferred to its facilities after the inauguration.
The transition official spoke on condition of anonymity because the person was not authorized to discuss post-election decisions publicly. They are building a case for firing Mueller.More news: Fowler back in the thick of it at US Open
Despite high public interest in the content of the memos, Comey's testimony that the records are not classified and a ruling from the Justice Department that the Federal Bureau of Investigation should expedite CNN's FOIA request for the memos, the Federal Bureau of Investigation has not provided either the documents or a reason to withhold them, according to the lawsuit filed in the US District Court for the District of Columbia.
The Democratic National Committee, in a statement Friday, said that Rosenstein, who has authority over Mueller, needs to recuse himself from the Russian Federation investigation and that control of the investigation should not be given to another presidential appointee. However, several days later, in an interview with NBC News's Lester Holt, Trump contradicted his administration and cited "this Russian Federation thing" as the reason for Comey's sacking. Rosenstein, who would be responsible for actually dismissing Mueller, told USA lawmakers he would fire him only with good cause.
The White House recently began referring all questions on Trump and Russian Federation to the lawyer Marc Kasowitz.
The order by the general counsel for his transition team casts a wide net on documents that could shed light on ties between Trump's presidential campaign and representatives of Russia's government.
Trump and his White House have offered various explanations for Comey's abrupt ouster, including that he lost the confidence of the bureau's rank and file - a suggestion flatly dismissed by acting director Andrew McCabe.
Trump has been growing increasingly agitated with Rosenstein over the last day or two in private White House meetings and in conversations with friends, multiple people have told CNN.
The Twitter attacks came as Vice President Mike Pence hired a personal lawyer to represent him in the intensifying investigation. Cohen has worked for Trump since the mid-2000s and was active in the campaign.