Editor's note: This story has been updated throughout.

WASHINGTON – The two senators from Texas said Wednesday that President Donald Trump was within the bounds of his authority to fire now-former FBI Director James Comey, even as the move raised concerns from some of their fellow Republicans and virtually all Democrats in the Senate.

Both U.S. Sens. John Cornyn and Ted Cruz will play important roles in choosing Comey's successor, both in their roles on the Senate floor and from their perches on the Senate Judiciary Committee, which oversees the Department of Justice.

Right after morning votes on Wednesday, Cornyn repeatedly told reporters that Trump was "within his authority" to fire Comey.

Cornyn insisted that the termination – which shocked Washington on Tuesday afternoon - would not change the investigation into Russian meddling in the 2016 presidential and Congressional elections.

"I've heard what I think is a phony narrative that he did this somehow to squelch the investigation into Russia, which I don't believe there's any evidence of," he said. "But if you assume that, this strikes me as a lousy way to do it. All it does is heightens the attention given to the issue."

"And we're going to conduct our investigation in the intelligence committee on a bipartisan basis," added Cornyn, the lone Texan on the Senate Intelligence Committee.

Cruz also backed Trump's firing of Comey in a written statement.

“The Director of the FBI needs to be above reproach, with an unquestioned reputation for fairness and impartiality," Cruz said. "Unfortunately, Mr. Comey had lost the confidence of both Republicans and Democrats, and, frankly, the American people."

"The next Director needs to be someone of the utmost integrity who can successfully restore the public’s confidence and lead the men and women of the FBI who selflessly serve and defend our great nation.”

The fear among many Democrats – and some Republicans – is that Trump will appoint an FBI director who will not be viewed as independent as he or she is tasked with investigating associates of the president.

Cornyn brushed off those concerns, arguing that the firing only brought more attention to the investigations.

"This doesn't change anything. The Department of Justice is going to continue its investigation," he said. "We're going to get a new FBI director, and Congress can judge for ourselves what we think about the integrity and independence of that person."

Cornyn, the second-ranking Senate Republican, said he did not receive a heads-up from the White House before news of Comey's firing broke but said that was "not surprising."

Most Senate Democrats as well as some of the chamber's Republicans have called for an independent prosecutor to investigate links between the Trump campaign and Russia.

When asked if he, Cornyn, might be interested in serving as FBI director, the senior senator from Texas demurred: "I'm happy serving my state and my country."

Texas House Republicans have been mostly silent on the matter, as they have been on most issues related to the investigation into Russian interference in the 2016 presidential and Congressional elections. The U.S. House of Representatives is on recess this week.

U.S. Rep. Will Hurd, a former CIA operative and member of the House Permanent Select Intelligence Committee, was the first Texas Republican to put out a critical statement of Trump's actions Wednesday.

"While I understand the desire to maintain the public trust in the FBI, the timing of James Comey's firing is troubling, especially since the Justice Department Inspector General was conducting an independent investigation into Comey's handling of the Clinton email case," Hurd, of Helotes, said in a statement. "Due to the ongoing criminal investigation into Russian interference in our elections, I'm concerned that Comey's firing is going to create new concerns around the public trust in the Department of Justice."

Hurd urged Trump to nominate a new FBI Director and added, "While Director Comey and I have disagreed in the past, I have always maintained and will continue to have the utmost respect for his public service, and most importantly his commitment to protecting this country."

Several House Democrats from Texas also put out statements critical of Trump's move. U.S. Rep. Joaquin Castro, a San Antonio Democrat who also sits on the House Permanent Select Committee on Intelligence, called Comey's firing "extremely suspicious" and that "the FBI’s investigation into Russia’s interference in the 2016 election must continue."

“Now more than ever, it is essential an independent commission is formed to investigate Russia’s interference in our election,” Castro said.

Once the House returns from its current break, though, U.S. Rep. Mike Conaway, R-Midland, is sure to draw strong scrutiny. Last month, he was tapped to lead the House Intelligence Committee investigation into Russian influence over the 2016 election after his predecessor, U.S. Rep. Devin Nunes of California, drew criticism following a series of unorthodox meetings and communications with White House officials.

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