The post Mr. Trump wrote on Monday took his criticism of the Justice Department to the next step, suggesting that defending the Republican majority in the House should determine whether two members are prosecuted.

“Two long running, Obama era, investigations of two very popular Republican Congressmen were brought to a well publicized charge, just ahead of the Mid-Terms, by the Jeff Sessions Justice Department,” he wrote. “Two easy wins now in doubt because there is not enough time. Good job Jeff.”

He was presumably referring to indictments of Representatives Duncan Hunter of California, who was charged with using more than $250,000 in campaign funds for personal expenses, and Chris Collins of New York, who was charged with insider trading. Both have pleaded not guilty.

Mr. Trump’s suggestion would have been a major scandal under any other president, veterans of past administrations said. “His interference in an ongoing criminal investigation may be the single most shocking thing he’s done as president,” said Walter E. Dellinger, a former acting solicitor general under President Bill Clinton.

Senator Jeff Flake of Arizona, a Republican who has been among the president’s most outspoken critics in his own party, had the same reaction. “Those who study this kind of thing say it’s a lot more evidence for abuse of power or obstruction,” he said. “I just know it’s not healthy for the institutions of government to have the president want to use the Department of Justice that way.”

Senator Lisa Murkowski, Republican of Alaska, likewise criticized the president’s comments. “I’m looking at them just as you are looking at them,” she told reporters. “I thought that yesterday’s comments were not appropriate and they upset me.”

Asked if Mr. Trump was only feeding Democrats’ interest in impeachment, she said: “I have no idea what he is doing. I have no idea what he is thinking.”