WASHINGTON — President Trump praised Gina Haspel on Monday as she was sworn in as the director of the Central Intelligence Agency, congratulating her on becoming the first woman to lead what he called “the most elite intelligence professionals on the planet.”

But even as Ms. Haspel took over as C.I.A. director, conservatives and some Trump allies were accusing her of being part of the “deep state” conspiracy that the president repeatedly claims has been conducting a “witch hunt” against him.

Publicly and privately, Trump supporters have been raising questions about Ms. Haspel’s loyalty to the president, and urging — without providing any evidence — an examination of what she knew about the intelligence community’s efforts to connect Trump to Russia.

In particular, they have questioned whether Ms. Haspel, a 33-year veteran of the spy agency who was the C.I.A.’s station chief in London, knew of the F.B.I.’s highly secretive interview of an Australian diplomat in London, and was aware that the bureau used an informant to gather information there from Trump associates about possible Russian coordination with Mr. Trump’s presidential campaign.