Trump was willing to move past Romney's bitter campaign-trail criticism of him. And incoming chief of staff Reince Priebus, who chaired the Republican National Committee during both men's presidential campaigns, convinced Trump there was value in showing he was big enough to bring Romney into the fold.
But top aides, including Kellyanne Conway and Steve Bannon, saw Romney's attacks as unforgivable, according to three sources close to Romney, two of whom are familiar with Trump's transition process.
CNN has learned that Vice President-elect Mike Pence and Priebus also wanted Romney to apologize as a way to smooth over the tension.
Trump personally saw it in business deal terms: He would get the mea culpa he sought from Romney; Romney would get the job he covets.
But Romney -- who titled his own book "No Apology" -- declined.
He offered forward-looking praise for Trump -- starting with the President-elect's election-night speech. But he wouldn't go backward and retract his words from the campaign.
Eventually, Romney's critics persuaded Trump to look elsewhere, and on Monday he tapped ExxonMobil CEO Rex Tillerson
as his nominee for secretary of state.
It was the culmination of a selection process that looked like an episode of "Celebrity Apprentice" -- with Romney, Tillerson, former New York City Mayor Rudy Giuliani, former Gen. David Petraeus and Senate Foreign Relations Committee Chairman Bob Corker being paraded in and out of Trump Tower as the President-elect narrowed his list.
Romney talks start after election
The conversations between Trump and Romney began after the former Massachusetts governor called Trump to offer his congratulations.
Trump missed the call. Pence called back -- and then Trump did, as well. Trump said Romney should come in to discuss a position in the Cabinet, a source said. In Romney's conversation with Pence he said the only position he was interested in was the top diplomat job.
Some Republicans saw the conversations as Trump forcing a former foe to grovel -- a move that would also undercut Romney as a future critic.
But Romney "never felt he was being played," a source said.
He had told friends for years that he wanted to serve again, and the only job he was interested in was secretary of state. He felt he was under "serious" consideration from Trump as the two met twice in person and spoke several times by phone.
"I have a hard time believing that Trump expended all this energy and time to simply tweak Mitt, and that Pence, Reince and others in the President-elect's inner circle would play along," said Dan Senor, a 2012 Romney adviser who remains in close contact with him. "The press was distracted by the theater of the public process, but behind the scenes there were substantive conversations and meetings, which reflected a serious hands-on process by the President-elect and his team; it was impressive."
Trump -- who has a history of striking business deals with one-time opponents -- was "pleasantly surprised at how well the two men got along; was willing to let bygones be bygones," a source said. He told others he was fond of Romney.
"Mitt made it hard on him because when they talked he really liked him," one source said. Trump thought Romney was smart and "looked the part" of America's top diplomat.
One source close to Romney said it was clear Trump was serious about considering him. Even when the selection process was dragging, the source said, Trump would call Romney to check in.
"The idea that there was some grand design to humiliate him is ludicrous," the source said.
Trump's inner circle pushes for apology
A turning point came when Conway went on television to blast Romney.
After that, more than once, Trump raised the prospect of a public apology from Romney -- who in a speech in Utah had called Trump "a con man, a fake and a phony"
who was "playing us for suckers," and had afterward attacked Trump on Twitter and courted Republicans to launch a third-party campaign.
"It was clearly a sticking point and bothered Trump," a source said.
The sources believe, though, that Trump wasn't asking for himself. He wanted an apology from Romney to soothe over tensions within his own camp.
Still, loyalty to Trump hasn't led to jobs in every case. Giuliani, New Jersey Gov. Chris Christie and former House Speaker Newt Gingrich were Trump's top surrogates and close advisers during the campaign, and none have received the posts they sought.
Not the only sticking point
Trump and Romney also faced another major divide: Russia.
The two men have polar opposite views: Romney believes Russian President Vladimir Putin cannot be trusted
or negotiated with; Trump strongly disagrees and feels he can make deals with Putin, the sources said.
The two were also divided on the United States' approach to NATO and the conflict in Syria, sources said.
"Mitt hoped he could balance Trump out on these issues. In the end he knew it would be Trump's decision, but he thought he could be an important voice on the team to weigh in and balance him out on these particular issues," one source said.
Romney and Trump never came to terms over Russia. And in Tillerson, who is friendly with Putin, Trump found someone closer to his worldview.
There was one more factor weighing on Trump's mind as he considered Romney: The former Massachusetts governor and 2012 GOP nominee was, as one source put it, "an independent player with his own power base" who Trump couldn't control.
If the two disagreed, Romney could leave the administration and criticize Trump publicly. Just as Romney hadn't helped Trump win the White House, Trump wouldn't have elevated Romney's already-prominent national stature.
"He didn't owe Trump," the source said.
'He does not regret it at all'
Senor said Romney doesn't regret going through Trump's selection process, even though he didn't get the secretary of state's post.
"He does not regret it at all," he said. "Romney has had his differences with Trump, but now Trump is going to be president. And Mitt genuinely believes that if he could help the president and serve the country, then he has an obligation to do so."
"Every decision that Mitt has made during this process has been public-spirited," he added. "The political culture can at times leave us all evaluating the motives of politicians through a very cynical lens. But Romney was prepared to subject himself to this process because of the country he loves and that he wants to serve. It's really that simple."