Mr. Cummings, speaking on ABC’s “This Week,” said that “there are probably many” violations by Mr. Trump. He also vowed to look into reports that Mr. Trump had blocked the federal government from moving the F.B.I. headquarters, which sits across Pennsylvania Avenue from his downtown Washington hotel, because doing so would have cleared the way for construction of a competing hotel.

“We’ve got to figure out when is he acting on behalf of the American people in a lot of his decisions or — or is he acting on his own behalf?” Mr. Cummings said. But he said he saw subpoenas as a last resort, and ducked a question about what he would do if the White House refused to comply with them.

“We will cross that bridge when we get to it,” he said.

House Democrats also plan to investigate whether Mr. Trump used what Mr. Schiff called “instruments of state power” to try to punish The Washington Post and CNN, whose journalists asked questions of the president that Mr. Trump did not like. The White House revoked the credentials of the CNN correspondent Jim Acosta last week after Mr. Acosta confronted Mr. Trump at a raucous news conference, and the president has warned that more journalists may lose their credentials.

Mr. Schiff, in an interview with the website Axios, said Congress should examine whether Mr. Trump attempted to block AT&T’s merger with Time Warner as “an effort to punish CNN.” He also accused Mr. Trump of “secretly meeting” with the postmaster general to prod her into “raising postal rates on Amazon,” whose chief executive and founder, Jeff Bezos, owns The Washington Post.

“This appears to be an effort by the president to use the instruments of state power to punish Jeff Bezos and The Washington Post,” Mr. Schiff said.

But perhaps nothing is as high on Democrats’ agenda as protecting the Mueller investigation. Mr. Whitaker has stirred deep concerns among Democrats since Mr. Trump, on the heels of the Republicans’ loss of the House last week, named him acting attorney general after firing Jeff Sessions, who had long endured Mr. Trump’s wrath over the Russia inquiry. Among other comments, Mr. Whitaker had once declared that there was “no collusion” between the Trump campaign and Russia — a remark that prompted Democrats to say he had prejudged the inquiry’s conclusion.

Mr. Whitaker’s appointment went outside the usual Justice Department plan of succession. Ordinarily, Rod J. Rosenstein, who had protected the Russia inquiry as deputy attorney general, would have ascended to the top job. Unlike Mr. Rosenstein, Mr. Whitaker, who was Mr. Sessions’s chief of staff, lacks Senate confirmation. Democrats say the appointment is unconstitutional.