WASHINGTON — Internal Republican divisions over the fate of Gina Haspel’s tenure as C.I.A. director have come tumbling into view as some Senate leaders showed support while President Trump’s allies pushed for her ouster, in part over the agency’s role in disseminating the whistle-blower complaint that prompted impeachment, according to current and former administration officials.

For weeks, Mr. Trump has been mulling whether to fire Ms. Haspel, the agency’s first female director. Despite Mr. Trump’s refusal to accept his election loss, people close to him understand that his time in office is limited and the window to remove her is dwindling.

Some officials and presidential allies believe that Ms. Haspel failed to do enough to stop the whistle-blower’s complaints about Mr. Trump’s July 2019 call with President Volodymyr Zelensky of Ukraine, which prompted impeachment, from moving forward. Others have also grown frustrated with her opposition to declassifying documents related to Russia’s 2016 election interference.

White House aides are divided over Ms. Haspel’s removal. Pat A. Cipollone, the White House counsel, opposes it, though other officials are pushing for her dismissal, the officials said. Allies of the president have had “a real issue of trust” with Ms. Haspel for more than a year, a senior administration official said.

The C.I.A. declined to comment.

Though Ms. Haspel had no direct role in the impeachment inquiry, it was prompted by a C.I.A. officer who made an anonymous and indirect complaint to the agency’s general counsel, then filed a whistle-blower complaint to the inspector general of the intelligence community.

Mr. Trump has acted against people he has perceived as aiding the impeachment inquiry. After the Senate acquitted the president this year, he fired Gordon D. Sondland, the ambassador to the European Union who testified in the inquiry, and Michael K. Atkinson, the inspector general who investigated the whistle-blower complaint. The White House also effectively blocked the promotion of Lt. Col. Alexander S. Vindman, the National Security Council aide who was the primary witness in the impeachment hearing, leading to his retirement.

White House officials who favor Ms. Haspel’s ouster believe she has been insubordinate to John Ratcliffe, the director of national intelligence, and boxed in the White House on the debate over whether to declassify Russia documents by sharing her concerns with Congress, the senior official said.

Ms. Haspel’s congressional allies noted that she is a cabinet official and has a responsibility to answer questions from lawmakers conducting oversight of the intelligence agencies.

Tensions over Ms. Haspel’s fate intensified this week after Mr. Trump ordered a string of firings at the Pentagon that began with the ouster of Defense Secretary Mark T. Esper on Monday and continued on Tuesday with the removal of other key Pentagon officials, including the head of intelligence and the leader of the influential policy apparatus. The White House then installed loyalists in top defense posts.

Top Republicans came subtly to Ms. Haspel’s defense on Tuesday. Senator Mitch McConnell invited her for a conspicuous meeting in his office.

That prompted Arthur Schwartz, an informal adviser to Donald Trump Jr., to accuse top Republicans of trying to manipulate Mr. Trump into keeping Ms. Haspel, who Mr. Schwartz said “undermines Trump and subverts his agenda at every turn.” Making reference to Ms. Haspel’s previous clandestine work overseas managing C.I.A. informants, Mr. Schwartz said Republicans “are getting played by a master case officer.”

Senator John Cornyn of Texas responded on Twitter, saying intelligence should not be partisan and is “about preserving impartial, nonpartisan information necessary to inform policy makers.”

That invited a response from Donald Trump Jr., who asked whether the senators had discussed Ms. Haspel with members of the administration. “Or,” he concluded in an apparent swipe at Ms. Haspel, “are you just taking a trained liar’s word for it on everything?”

Some of Mr. Trump’s allies believe the Russia documents they wanted declassified include information that will undermine established facts about the Trump campaign’s ties to Russia. But others are skeptical the documents will change the understanding of what happened in 2016. Outside investigators who reviewed intelligence gathered at the time for a bipartisan Senate report released this year backed the conclusion that Russia favored Mr. Trump in that election.

No decision on whether to declassify the documents has been made. But the ultimate authority to declassify them does not rest with Ms. Haspel. Mr. Ratcliffe or Mr. Trump could ultimately decide to release them, with or without Ms. Haspel’s blessing.