According to an unnamed source who spoke with the Washington Post, Donald Trump asked Mitt Romney to publicly apologize for his critical comments, but Romney refused.

On Monday, Donald Trump declared Exxon Mobil executive Rex Tillerson as his pick for secretary of state, passing on the opportunity to name Mitt Romney as a candidate. 
The decision did not come as a surprise to many as Romney and Trump have had a troubled past. 
According to an unnamed source who spoke with the Washington Post, the situation may have been a bit more complicated. 
That individual, who is described as a “close Romney ally,” revealed that Trump asked the 2012 presidential candidate to publicly apologize for making unflattering remarks during the recent campaign, but Romney refused. 
Among the comments in question was one made in March, alleging, “Donald Trump is a phony, a fraud. His promises are as worthless as a degree from Trump University.”
According to CNN, Trump was not alone in wanting Romney to retract such statements.
Reince Priebus and Mike Pence also pushed for the public expression of regret.
Mitt Romney appears to be handling the situation well. On Monday he wrote on Facebook, “It was an honor to have been considered for Secretary of State of our great country. My discussions with President-elect Trump have been both enjoyable and enlightening. I have very high hopes that the new administration will lead the nation to greater strength, prosperity and peace.”