So many underlings doing their jobs, so much disloyalty.

Mr. Trump first expressed his dismay about Michael Atkinson around the time the whistle-blower’s complaint became public in September. In recent weeks, he has continued to raise with aides the possibility of firing him, one of the people said.

The president has said he does not understand why Mr. Atkinson shared the complaint, which outlined how Mr. Trump asked the Ukrainian president to investigate Mr. Trump’s political rivals at the same time he was withholding military aid from the country. He has said he believes Mr. Atkinson, whom he appointed in 2017, has been disloyal, one of the people said.

Mr. Trump’s private complaints about Mr. Atkinson have come as he has publicly questioned his integrity and accused him of working with the Democrats to sabotage his presidency.