Mr. Gingrich added that he had told Mr. Trump that a Trump-Gingrich combination would be a bolder pairing of "two pirates" on the same ticket.
Selecting Mr. Pence, by contrast, might ease relations between Mr. Trump and Republicans in Washington, where party leaders have eyed his every move with grave apprehension.
Republicans on Capitol Hill spoke approvingly of Mr. Pence on Thursday: He is seen among his former colleagues there as a conventional politician with standard-issue conservative beliefs, including on some subjects where his policy instincts plainly conflict with Mr. Trump's.
Mr. Pence has endorsed free-trade agreements, including the Trans-Pacific Partnership, an Asian trade deal that Mr. Trump has described as a "rape" of the American economy. As a House member, Mr. Pence also voted for the Iraq War, which Trump has condemned, and last winter he denounced Mr. Trump's call to ban all Muslim immigration into the United States.
If those views place Mr. Pence at odds with Mr. Trump, they are in line with the outlook of Republican leaders in Congress.
"It's no secret I'm a big fan of Mike Pence's," said Paul D. Ryan, the speaker of the House from Wisconsin. "We're very good friends. I have very high regard for him. I hope that he picks a good movement conservative. Clearly, Mike is one of those."
The Senate majority leader, Mitch McConnell, of Kentucky, said picking Mr. Pence would be a "good move by Donald Trump." Mr. McConnell, who has sharply rebuked Mr. Trump in recent weeks for his indiscipline on the campaign trail, said he would "look forward to enthusiastically supporting the ticket."