WASHINGTON — President Trump said on Monday that the White House was “trying to find out” the identity of the whistle-blower whose claims led Democrats to begin an impeachment inquiry last week, even as the whistle-blower’s lawyers have outlined “serious” safety concerns for their client as Mr. Trump has repeatedly targeted him and compared him to a spy.

Mr. Trump’s latest comment, made to reporters in the Oval Office during the swearing-in of his new labor secretary, Eugene Scalia, followed up on a series of Twitter posts over the weekend, in which Mr. Trump claimed that he deserved “to meet my accuser.”

It was not immediately clear what steps the White House was taking to identify the whistle-blower, but the White House has known for weeks that a C.I.A. officer lodged concerns about Mr. Trump’s dealings with Ukraine. Still, Mr. Trump’s fixation on discovering and discussing the identity of the whistle-blower, whose anonymity is protected by law, was seen as a brazen move for a president under scrutiny for abuse of power.

“As the acting D.N.I. testified last week, the law and policy supports protection of the identity of the whistle-blower from disclosure and from retaliation,” Mark Zaid, the lawyer for the whistle-blower, said Monday, referring to the acting director of national intelligence, in response to Mr. Trump’s most recent comments. “No exceptions exist for any individual.”