Updated at 7:30 p.m. with comments from former FBI director James Comey

WASHINGTON -- As President Donald Trump defended his decision to oust FBI director James Comey, a few key Republicans have joined Democrats in raising concerns about the timing of his abrupt dismissal in light of the agency's probe into the Trump campaign's ties with Russia.

But Texas Sens. Ted Cruz and John Cornyn aren't among them.

In a statement released Wednesday, Cruz agreed with the president's decision, saying Comey has "lost the confidence of both Republicans and Democrats," while Cornyn took to Twitter to slam Democrats as hypocrites for decrying the firing of a man they've also assailed.

Their positions are in contrast to Senate Intelligence Chairman Richard Burr, R-N.C., Arizona Sens. John McCain and Jeff Flake and others who say they're troubled by the news. Burr's committee is leading the Senate probe into Russia's involvement in the election.

“I have found Director Comey to be a public servant of the highest order,” Burr tweeted on Tuesday, later adding: “His dismissal further confuses an already difficult investigation by the Committee."

His dismissal, I believe, is a loss for the Bureau and the nation. — Richard Burr (@SenatorBurr) May 9, 2017

Flake, a member of the Senate's judiciary and foreign affairs panel, tweeted that he's spent "several hours trying to find an acceptable rationale for the timing of Comey's firing. I just can't do it."

I've spent the last several hours trying to find an acceptable rationale for the timing of Comey's firing. I just can't do it. — Jeff Flake (@JeffFlake) May 10, 2017

The White House says Comey's firing was necessary to restore the public's confidence in the agency following his controversial handling of an investigation into Hillary Clinton's use of a private email server. At the White House on Wednesday, Trump only said, "He wasn't doing a good job. Very simply. He was not doing a good job."

Attorney General Jeff Sessions and newly tapped deputy attorney general, Rod Rosenstein, pushed for Comey's dismissal, according to the White House.

But the timing of his firing -- as well as Trump's dismissal letter to Comey -- is raising questions about whether Trump's administration sought to oust the man leading the investigation into whether his campaign colluded with Russia to sway the 2016 election.

In his termination letter, Trump thanked Comey for informing him, "on three separate occasions" that he is "not under investigation."

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Democrats -- as well as McCain and a handful of House Republicans -- are now renewing the call for an independent investigation. That includes Virginia Sen. Mark Warner, the leading Democrat on the Intelligence Committee, who said Comey's firing "demands the appointment of a Special Counsel."

.@POTUS firing #FBIDirector Comey in the middle of his investigation into the Trump campaign demands the appointment of a Special Counsel. — Mark Warner (@MarkWarner) May 10, 2017

Removal of Director Comey only confirms need for select cmte to investigate #Russia's interference in 2016 election https://t.co/LfKlwSw6iQ — John McCain (@SenJohnMcCain) May 9, 2017

Until now, most GOP leaders have resisted that push.

Cruz didn't address whether the Senate should pursue an independent investigation in his statement Wednesday but said "the next director needs to be someone of utmost integrity who can successfully restore the public's confidence."

Speaking to reporters later, he acknowledged that the timing was less than ideal. "It would have been clearly better for the decision to have been made in the opening week of the administration," he said.

Comey is the latest Trump official to be ousted. In February, the president booted former National Security Adviser Michael Flynn amid reports that he misled Vice President Mike Pence about his communications with Russian officials. He also dismissed Sally Yates, an Obama holdover who briefly served as his acting attorney general.

In an interview with Fox News, Cornyn, the No. 2 Republican in the chamber, dismissed concerns that Comey's firing jeopardizes ongoing investigations into Russian election meddling.

"Nothing will change by virtue of Director Comey's termination," said Cornyn, who sits on the Senate Intelligence Committee handling one of the many inquiries.

"We have a bipartisan commitment to get to the bottom of this Russian investigation," Cornyn said, adding that Rosenstein, the deputy, is "committed to getting to the bottom of the Department of Justice investigation, as well."

In the hours following Comey's dismissal on Tuesday, Cornyn took to social media to criticize the Democrats criticizing the firing. Last fall, Democrats universally panned the FBI director for reigniting concerns over Clinton’s use of a private email server just weeks before the election.

“Ds were against Comey before they were for him,” he wrote late Tuesday.

Ds were against Comey before they were for him. — Senator John Cornyn (@JohnCornyn) May 10, 2017

He was joined by South Carolina Sen. Lindsey Graham, who also downplayed reports that the firing will derail investigations.

"The FBI is not Mr. Comey. ... If you think the FBI will stop their investigation, it won't," Graham told MSNBC's Morning Joe. "But you've got to understand the Democrats have called for this guy to be dismissed for months. ... This gives hypocrisy a bad name."

The House is on recess this week, but many members have weighed in from their districts.

Texas reaction

Dallas Rep. Pete Sessions, whose father, Williams Sessions, was ousted as FBI director by President Bill Clinton, said he favors "dialogue over firing" and that Comey should have been allowed "to exit in a more gracious way."

"I think you treat good people better," Sessions told The Dallas Morning News' editorial page editor.

Rep. Will Hurd, R-San Antonio, said in a statement that the timing is "troubling" and could "create new concerns around the public trust in the Department of Justice." But he stopped short of endorsing an independent investigation.

Heath Republican Rep. John Ratcliffe, a former U.S. attorney, told Fox Business that while he likes Comey, the dismissal was "appropriate." Ratcliffe then floated South Carolina Republican Rep. Trey Gowdy as a potential replacement.

"We have survived bad presidents before, Republicans and Democrats," Ratcliffe said. "What the republic can't survive is if the people lose faith and trust in agencies like the FBI and the Department of Justice to fairly investigate and prosecute violations of the law ... and on both sides of the aisle, people have been levying that complaint."

Plenty of Texas Democrats greeted the news with skepticism, if not outrage.

San Antonio Democrat Joaquin Castro, a member of the House Intelligence Committee, called the firing "very suspicious." In a press conference on Wednesday, Houston Rep. Al Green said Trump should be investigated for obstruction of justice and that, if guilty, he should be impeached, according to Houston Public Media.

Rep. Joaquin Castro says the firing of FBI Director James Comey was "very suspicious" https://t.co/tgYGgsBCjQ https://t.co/ThOPJssWKw — CNN (@CNN) May 10, 2017

A besieged Trump unleashed a flood of tweets attacking Democrats, especially leading critic Sen. Richard Blumenthal of Connecticut.

Cryin' Chuck Schumer stated recently, "I do not have confidence in him (James Comey) any longer." Then acts so indignant. #draintheswamp — Donald J. Trump (@realDonaldTrump) May 10, 2017

The Democrats have said some of the worst things about James Comey, including the fact that he should be fired, but now they play so sad! — Donald J. Trump (@realDonaldTrump) May 10, 2017

James Comey will be replaced by someone who will do a far better job, bringing back the spirit and prestige of the FBI. — Donald J. Trump (@realDonaldTrump) May 10, 2017

Money for Russia investigation

The Justice Department also is flatly dismissing reports that, in the days before his termination, Comey requested more money and personnel for the bureau's investigation into Russia's interference.

Comey had been scheduled to appear before Burr's panel, along with other national security officials, to broadly discuss "worldwide threats" on Thursday.

He'll be replaced by Acting Director Andrew McCabe, the committee announced, but has been invited to testify before the same panel next week.

In a farewell letter obtained by CNN late Wednesday, Comey said he's "long believed that a president can fire an FBI Director for any reason, or for no reason at all."

He urged his staff to remain focused on the mission.

"In times of turbulence, the American people should see the FBI as a rock of competence, honesty, and independence," he wrote, later adding: "My hope is that you will continue to live our values and the mission of protecting the American people and upholding the Constitution."

Staff writer Jordan Rudner contributed to this report.