Trump did not answer questions about Flynn when asked by pool reporters at an education listening session on Tuesday morning.
Mr Trump is said to favour Vice Admiral Robert Harward, a former Navy SEAL, according to a White House official.
"The FBI must accelerate its investigation of the Russian connection with the Trump administration, and Congress must call for a bipartisan, independent, outside commission to fully investigate Russia's influence on the administration and the election", House Minority Leader Nancy Pelosi, D-Calif., said.
Yet now Trump is fuming, blasting the US intelligence community, accusing them of leaking classified information about former national security adviser Michael Flynn, who resigned Monday amid reports he had misled officials about his contacts with Russian Federation.
Flynn stepped down after leaks surfaced earlier this week of him discussing U.S. sanctions against Russia during a phone conversation with Russian ambassador to Washington Sergey Kislyak.
U.S. intelligence agencies previously concluded that Russian Federation hacked and leaked Democratic emails during the presidential campaign as part of efforts to tilt the vote in the Nov 8 election in Mr Trump's favour.
The conversation took place before Gen Flynn took up his NSA post and also before Mr Trump was inaugurated as president. It was strictly coincidental that they had a conversation.
Flynn acknowledged in his February 14 resignation letter that he misled Pence.
On Thursday, Trump vowed to punish those leaking classified information to the press, including the details about phone calls between former National Security Advisor Michael Flynn and the Russian ambassador.
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People familiar with the agency's multiple probes said there was no evidence so far of pre-election collusion between Russians and Mr Trump's campaign, or any evidence of criminal activity by Mr Flynn or anyone else connected to the President.
"Either Trump hasn't found the necessary independence and he's been driven into a corner. or "Russophobia" has permeated the new administration from top to bottom", he said.
Trump, Feb. 15: General Flynn is a wonderful man. The president declined to address that revelation and, as he has at other times in recent days, took questions at his news conference only from conservative news organisations and ignored more challenging questions shouted to him as he left the podium. I think it should be straight.
The characterization of the media's handling of the Flynn saga as "fake" is, well, fake.
Flynn had been incorrect on both counts and with the senior-most officials in the administration, including Vice-President Mike Pence, who were fielded by the White House to defend the embattled NSA in interviews and interactions on all major television news networks. That is why Flynn's resignation was unavoidable.
Senate Democrats have called for a "full classified briefing by all relevant agencies, including the Department of Justice and the Federal Bureau of Investigation, as soon as possible and certainly before Thursday, February 16". Asked to confirm that "the sanctions that Flynn was discussing were the sanctions for the election hacking", Spicer replied, "Right". At the time, Trump said he "didn't know about it". "We don't even know who listened to the phone calls of an American citizen yet".
Here is what happened: After the Post story broke late Feb. 9, other news outlets quickly followed with stories of their own on Feb. 9 and Feb. 10.
Mr Flynn was also reportedly questioned by Federal Bureau of Investigation agents in his first days as national security adviser, according to U.S. media. What report is that? "I'll look at that". How do we know? Less than an hour later, Spicer issued a statement saying that Trump was "evaluating the situation".
The Kremlin earlier said that it was not expecting a breakthrough before the two presidents meet in person.
Today, Ryan said, "I'm not going to prejudge the circumstances surrounding this".