Asked about Mr. Trump’s deliberations, a spokesman, Jason Miller, said in an email Thursday, “The president-elect is meeting with a number of well-qualified potential selections for this important position who share his America First foreign policy — some of whom have been made public and others who have not — and the president-elect will make public his decision when he has finalized it.”

Mr. Romney would represent a departure from the hard-liners Mr. Trump has already picked for his national security team. But aides like Stephen K. Bannon, Mr. Trump’s chief strategist, have expressed doubts about Mr. Romney’s loyalty given his denunciation of Mr. Trump as a “phony” and a “fraud.” Mr. Bannon and others have told colleagues they fear that a State Department under Mr. Romney could turn into something of a rogue agency.

Asked to explain her Twitter post about Mr. Romney, Ms. Conway said that while she trusted Mr. Trump’s judgment, she found it notable that the most outrage directed at Mr. Trump from the party’s grass-roots “is not against something he said, but something he may do.” In another post, she said that being “loyal” was an important characteristic for a secretary of state.

Others hoping to catch Mr. Trump’s ear have taken their message to a place they know he is likely to absorb it: cable news. Joe Scarborough, the MSNBC host, who has spoken with Mr. Trump about his concerns that Mr. Giuliani would not be confirmed by the Senate, has taken to making those arguments on a daily basis on his morning show, which he knows Mr. Trump watches.

Others, like Newt Gingrich, the former Republican House speaker, and Mike Huckabee, the former governor of Arkansas, have gone on television to try to dissuade Mr. Trump from picking Mr. Romney. Mr. Huckabee, who said during the 2008 presidential campaign that Mr. Romney reminded voters of “the guy who laid them off,” told Fox News on Wednesday that picking Mr. Romney would be “a real insult” to Mr. Trump’s supporters. Mr. Giuliani is a favorite of the Republican voters who turned out in large numbers to lift Mr. Trump to victory.

Sean Hannity, a Fox News host whose opinion Mr. Trump often privately solicits, has also been deeply critical of Mr. Romney on his show.