It was unclear whether the president's comment confirming he was under investigation was based on direct knowledge or media reports by NBC News and others that Mueller is examining whether the president obstructed justice by firing Comey.
ABC said Rosenstein told Associate Attorney General Rachel Brand she would have authority over the probe if he were to step aside.
Rosenstein appointed special counsel Robert Mueller to investigate alleged Russian meddling in the 2016 USA election, and has told lawmakers he would only fire him with good cause.
Amid stirrings earlier this week that Trump was possibly considering the idea of firing Mueller, it was Rosenstein who testified before a congressional hearing that such an act could not be taken without specific cause.
Postscript: Rosenstein sent out a very odd press statement last night, urging Americans to be skeptical of reports based on anonymous sources.
Deomcratic Sen. Dick Durbin said Friday that President Donald Trump will put the country at the "brink of a constitutional crisis" if he fires special counsel Robert Mueller.
But any involvement of Rosenstein - who himself only took over the probe after Sessions recused himself - is complicated by the fact that he authored the memo that Trump cited in terminating Comey.
President Donald Trump speaks in the Roosevelt Room of the White House in Washington, Thursday, June 15, 2017, during an event on Apprenticeship and Workforce of Tomorrow initiatives. Earlier this week the Washington Post released a story reporting Mueller is investigating President Trump for obstruction of justice.
Making matters worse, Trump's tweet isn't altogether true, either. "Witch Hunt", the president wrote in his tweet.
North Korea fires several cruise missiles into East Sea: South Korean military
No one doubts that Kim Jong Un wants an ICBM; many, however, wonder whether his missileers are close to delivering one that works. Syring however assured U.S. lawmakers that America has the ability to defend itself against the North Korean threat.
In addition, "the officials who described the financial focus of the investigation spoke on condition of anonymity because they were not authorized to speak publicly".
Trump had previously said he was "going to fire [Comey] regardless of the recommendation" Rosenstein and several others sent him, suggesting he let go of the director, who was serving a 10-year contract as the head of the federal agency.
The Twitter attacks came as vice president Mike Pence hired a personal lawyer to represent him in the intensifying investigation.
Flynn, dismissed by Trump in February after it emerged he had misled Pence about conversations with the Russian ambassador, is a subject in investigations by intelligence committees in the House of Representatives and Senate, as well as the FBI.
There was already some discussion about whether Rosenstein would also have to recuse himself - he may be a witness to the president's alleged crimes - and Trump admonishing Rosenstein in public probably makes the DOJ official's recusal more likely.
"They made up a phony collusion with the Russians story, found zero proof, so now they go for obstruction of justice on the phony story".
Mr Trump also repeatedly tried to convince Mr Comey to publicly state that he was not being personally investigated in the Russian Federation probe, which he declined to do.
"Despite the phony Witch Hunt going on in America, the economic & jobs numbers are great".