CNN and NBC said Trump would meet over the weekend with Romney, the former governor of Massachusetts who was the Republicans' unsuccessful 2012 White House candidate against President Barack Obama.
NBC said that Trump was considering Romney as secretary of state, which would put a figure with more orthodox Republican views in charge of US foreign policy.
Senator Jeff Sessions, an arch-conservative Republican from Alabama who is one of Trump's closest allies in Congress, said he expected the incoming president to consider the "capable" Romney for some position.
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"I think it's good that the president-elect is meeting with people like Romney. There are a lot of talented people that he needs good relationships with," said Sessions, himself a top contender for a cabinet post.
"And I think Mr Romney would be quite capable of doing a number of things," Sessions told reporters after meeting the president-elect at his Trump Tower in Manhattan.
Romney was one of the staunchest Republican opponents of Trump's candidacy during the party's primaries, describing the businessman as vulgar, unprincipled and threatening to US values.
Romney in particular chastised Trump for proposing a ban on all foreign Muslims entering the United States.
In March, Romney said that "Trump's bombast is already alarming our allies and fueling the enmity of our enemies."
Romney also has a more traditional Republican skepticism of Russia, which he called the top geopolitical threat to the United States during the 2012 election.
In a striking departure for a Republican, Trump has voiced hope for working with Russian President Vladimir Putin, who has spoken warmly of the businessman-turned-world leader.
Media reports have speculated on a wide range of names to be secretary of state including South Carolina Governor Nikki Haley, another Republican who was initially lukewarm to Trump.
An Indian American, Haley would inject diversity into the cabinet after a divisive election in which Trump was outspoken in his criticism of immigration.
Other names floated for secretary of state include Rudy Giuliani, the combative former New York mayor and staunch Trump defender who would likely face scrutiny over a slew of business dealings, and hawkish former US ambassador to the United Nations John Bolton.
