If Sessions' departure was an "opening move" by Trump to meddle in Mueller's investigation, Hoyer said in a statement, "the president must be held accountable".
He called on Whitaker to recuse himself from the Russian Federation probe because of previous comments about the investigation, echoing calls from other top Democrats. The bill has been reported out by the Senate Judiciary Committee and King has previously voiced his backing of the bill on Twitter.
House Democratic leader Nancy Pelosi called it a "blatant attempt" to undermine the Russian Federation probe, while Senate Democratic leader Chuck Schumer tweeted that "clearly, the president has something to hide".
The special counsel's office declined to comment Wednesday after Sessions' resignation.
Clovis, the 2016 Trump campaign national co-chairman, said in a May radio interview that he has been interviewed by the special counsel's team and had testified before the Mueller grand jury.
Democrats raised concerns about Sessions' acting replacement, Matthew Whitaker, who now oversees Mueller and once argued Mueller's probe was going too far. This left the investigation in the hands of Deputy Attorney General Rod Rosenstein.
Whitaker has previously questioned the scope of the Mueller probe, writing an opinion column for CNN past year charging that the investigation would be going too far if expanded to include Trump family finances.
One is simply by declining to continue to pay the investigators or attorneys working for the special counsel.More news: Thugs of Hindostan movie review: Aamir Khan, Amitabh Bachchan - starrer disappoints
"If this wasn't about the Mueller probe, why install an Acting AG other than the DEPUTY Attorney General?" he wrote in a tweet.
And, if nothing else, having an attorney general who isn't recused from Mueller's work might give the White House a clearer look inside it.
"It's very significant because Whitaker's position on indictments or future indictments may be different than Rosenstein's, and Rosenstein had given Mueller a broad mandate to pursue various crimes", said Washington criminal defense lawyer Jeffrey Jacobovitz.
Trump's relentless attacks on Sessions came even though the Alabama Republican was the first US senator to endorse Trump and despite the fact his crime-fighting agenda and priorities, particularly his hawkish immigration enforcement policies, largely mirrored the president's. He recused himself the next day, saying it would be inappropriate to oversee an investigation into a campaign he was part of. He also announced media leak crackdowns, tougher policies against opioids and his Justice Department defended a since-abandoned administration policy that resulted in parents being separated from their children at the border. Mueller's team, which has interviewed Sessions, has been investigating the president's attacks on him and his demands to have a loyalist in charge of the Russian Federation investigation.
Sessions, who likely suspected his ouster was imminent, was spotted by reporters giving some of his grandchildren a tour of the White House over the weekend.
"We are immediately issuing multiple letters to key officials demanding that they preserve all relevant documents related to this action to make sure that the investigation and any evidence remains safe from improper interference or destruction", Nadler said.
Never in modern history has a president attacked a Cabinet member as frequently and harshly in public as Trump did Sessions, 71, who had been one of the first members of Congress to back his presidential campaign in 2015.