Amid a deepening crisis over the relationship between President Donald Trump's aides and Russia, some senior Republicans on Wednesday issued their boldest challenge yet and vowed to get to the bottom of the matter, while Democrats demanded an independent probe.

Trump, facing rising unease among fellow Republicans in Congress less than a month into his presidency, sought to focus the attention on what he called criminal intelligence leaks about his ousted national security adviser, calling Michael Flynn a "wonderful man" who was mistreated by the news media.

The New York Times reported on Tuesday that phone call records and intercepted calls showed members of Trump's presidential campaign and other Trump associates had repeated contacts with senior Russian intelligence officials in the year before the Nov. 8 election in which Trump defeated Democrat Hillary Clinton.

Republican Trump critics including Senators John McCain and Lindsey Graham voiced fresh consternation, but comments by Senate Foreign Relations Committee Chairman Bob Corker, who has been a Trump supporter, increased the pressure on the White House.

Corker said the Russia issue was threatening Trump's agenda on foreign affairs and domestic matters like healthcare and tax policy. He questioned whether the White House was able to stabilise itself and said Flynn should testify before Congress.

Democrats, doubting that either Trump's Justice Department or the Republican-led Congress will pursue the matter vigorously, demanded an independent investigation of possible illegal communications between Flynn and the Russian government and any efforts by Flynn or other White House officials to conceal wrongdoing.

The Democrats called for either a special counsel appointed by Trump's attorney general, Jeff Sessions, or the creation of a bipartisan commission with subpoena power. The top Senate Democrat, Chuck Schumer, said Sessions, a close ally of Trump, must recuse himself from any investigation.

Graham called for a broader bipartisan congressional investigation, to be conducted by a newly formed special committee rather than existing committees, if it turns out that Trump's presidential campaign communicated with the Russians.

"It's time for us to look into all things related to Russia's involvement in 2016," Graham told reporters, referring to last year's election.

But the top Republicans in the Senate and House of Representatives have insisted the matter be investigated by existing Republican-led committees.

U.S. intelligence agencies previously concluded that Russia hacked and leaked Democratic emails during the election campaign as part of efforts to tilt the vote in Trump's favour.

Some experts expressed concern the White House could curtail or divert probes into Flynn and Russian involvement in the election unless Congress becomes more aggressive by holding hearings and appointing an independent commission or special prosecutor into whether Trump's team violated federal laws in their contacts with Russia.

Intelligence agencies now overseen by Trump may not be ideally suited to the job, they added.

"It's not, at the end of the day, the job of the intelligence community to regulate the White House - and it shouldn't be," said Stephen Vladeck, a University of Texas law professor who focuses on constitutional law and national security.

Flynn was abruptly forced out by Trump on Monday after disclosures he had discussed U.S. sanctions on Russia with the Russian ambassador to the United States, before Trump took office, and that he had later misled Vice President Mike Pence about the conversations.

The Defense Intelligence Agency, the Pentagon spy agency once headed by Flynn, formally suspended his security clearance allowing him access to classified information, DIA spokesman James Kudla said.

The drama of Flynn's departure was the latest in a series of White House missteps and controversies since Trump was sworn in on Jan. 20.

"Let's get everything out as quickly as possible on this Russia issue," Corker told MSNBC's "Morning Joe" programme. "Maybe there's a problem that obviously goes much deeper than what we now suspect."

'VERY UN-AMERICAN'

In Twitter posts on Wednesday, Trump called the reported Russian connection with his campaign team nonsense, adding: "The real scandal here is that classified information is illegally given out by 'intelligence' like candy. Very un-American!"

At a news conference later with Israeli Prime Minister Benjamin Netanyahu, Trump charged that intelligence leaks to the news media were "a criminal act." He said Flynn, a retired U.S. Army lieutenant general, was "a wonderful man" who has been treated "very, very unfairly by the media."

But White House spokesman Sean Spicer said on Tuesday that Trump himself asked for Flynn's resignation because of the president's "eroding level of trust" in Flynn after a series of "questionable instances."

In his comments at the news conference, Trump did not address the accuracy of the material he said was being leaked.

Spicer on Tuesday also denied there had been any contact between any member of Trump's campaign team and Russia.

From early on in his White House bid, Trump said he would like improved relations with Russian President Vladimir Putin, a stance criticized by Democrats and also by some Republicans concerned about Washington softening its stance after Russia's annexation of Crimea from Ukraine and aggression in Syria.

Only a few Republican lawmakers have supported even the idea of extending any investigation to cover actions by Trump's team in the weeks after the election, when Flynn made his calls.

House Democratic leader Nancy Pelosi called on the FBI to expedite an investigation into the financial, personal and political ties of Trump and members of his administration to the Russians.

"There are suspicions that have arisen about the president of the United States," Pelosi said, including behaviour she called "very dangerous to the national security of our country" including poor judgement in appointing Flynn in the first place.

The Trump administration has offered Flynn's former job to U.S. Navy Vice Admiral Robert Harward, said two U.S. officials familiar with the matter. It was not immediately clear if Harward, a former deputy commander of U.S. Central Command, had accepted the offer, according to sources.

The Times, citing current and former U.S. officials, said U.S. law enforcement and intelligence agencies intercepted the communications around the same time they discovered Russia was trying to disrupt the presidential election by hacking into the Democratic National Committee.

Reuters could not immediately confirm the Times report, which the Kremlin dismissed as groundless. CNN also reported that Trump advisers were in constant contact with Russian officials during the campaign.