During the election campaign, Mr Romney called Mr Trump a "fraud" and "phony", while Mr Trump said Mr Romney`s unsuccessful campaign against Barack Obama in 2012 was "the worst ever".
Mr Romney later said Saturday`s talks were "far-reaching" and "in-depth".
Mr Trump has settled several posts so far, a number of them controversial.
The nominee for attorney general, Jeff Sessions, was rejected from becoming a federal judge in 1986 because of alleged racist remarks.
Lt Gen Michael Flynn, the new national security adviser, has drawn concern over his strident views on Islam.
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Separately on Friday, Mr Trump settled three lawsuits for fraud brought against him over his Trump University.
He tweeted on Saturday that he had settled "for a small fraction of the potential award because as President I have to focus on our country".
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