Trump denies any such links and says Comey had assured him he was not being investigated.
Related: What is a special prosecutor; who appoints one; what do they do?
On Friday, Trump raised eyebrows by tweeting a warning to Comey, whom he fired unexpectedly on Tuesday, about leaking information to the media, saying he "better hope that there are no "tapes" of our conversations before he starts leaking to the press".
Pressed by anchor Jake Tapper if he meant United States institutions were under assault internally from the President, Clapper responded, "Exactly".
Schumer also warned that Senate Democrats might refuse to vote on a new Federal Bureau of Investigation director until a special prosecutor is named to investigate alleged Russian meddling in the November 8, 2016, United States election.More news: Moms want dinner and quality time for Mother's Day
Already Mr Trump's decision appears to have emboldened the Senate intelligence committee investigating Russia's election interference and the president's associates, with lawmakers announcing a subpoena for former national security adviser Michael Flynn.
Graham also said Trump should forgo nominating a politician to be the new FBI director and instead promote someone from within the bureau.
Mr Spicer was absent from almost all press briefings in the days following the Comey firing.
Republican Senator Lindsey Graham said the White House needed to "clear the air" about whether tapes existed.
Warner also said he'd like to see Comey testify at a public session of his committee.More news: Myanmar, EU at odds over Rohingya rights mission
"The bottom line is I don't know if there was collusion, political collusion", Clapper told Tapper.
After Trump tweeted in March that President Barack Obama had ordered a wiretap on his phones in Trump Tower, three ex-employees told The Associated Press that as a businessman, Trump not only anxious about possible listening devices in his phones and office but had also occasionally taped his own phone conversations.
The interview on Fox comes after a week of intense scrutiny of Trump following his unexpected dismissal of Comey. Josh Holmes, a former aide to Senate Majority Leader Mitch McConnell, said later on the program that the Republican leader had just called him to say he would support Garland for FBI's top spot.
Warner said he would "absolutely" want the Senate to subpoena the White House to produce any recordings made by the Trump administration that are relevant to the Comey firing or the Russian Federation matter. Wallace said that on Saturday morning, the White House announced it would not make anyone available to discuss that topic - but did offer to book senior officials to talk about the president's upcoming foreign trip.
"I got to know a lot of them really well - and they are great people", he said. "And I think both should be non-partisan, not from either political party", Schumer said on CNN Sunday. If appointed and confirmed as FBI director, Garland's appointment would create a vacancy on the US Circuit Court of Appeals for the District of Columbia, which has jurisdiction over the DC area and federal agencies, making it one of the highest legal authorities in the country short of the Supreme Court. While Republicans control Congress, members of both parties have said they will ask tough questions about the nominee's independence.More news: 'Dr Phil' to air interview with Aaron Hernandez's fiancee
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