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Party leaders are normally eager to rally around a presidential nominee to combine forces for the battle leading up to November's general election. But Ryan has withheld his endorsement of Trump out of concern over the billionaire businessman's incendiary tone and policy ideas, which run counter to longtime Republican doctrine.
Ryan has opposed Trump's proposals to temporarily ban Muslims from entering the United States, deport 11 million illegal immigrants and impose protectionist trade policies.
In remarks to reporters after the meeting, the congressman stopped short of endorsing the 69-year-old New Yorker but said he was encouraged by the session.
"There's no secret that Donald Trump and I have had our differences. We talked about those differences today," Ryan said at his weekly news conference. "I do believe we are planting the seeds in getting ourselves unified."
Ryan, who may harbor aspirations of running for president in 2020 or later, noted that he represents a wing of the conservatives and that it is positive that Trump is bringing new voters into the party.
"The point, though, is: Can we agree on the common core principles that unite all of us?" Ryan said.
After their private session, mediated by Republican National Committee (RNC)Chairman Reince Priebus, Trump and Ryan then attended a wider meeting with other Republican leaders. Ryan, 46, of Wisconsin, is the nation's top elected Republican and is seen as a leader of the party establishment that has resisted Trump's candidacy.
Outside the RNC headquarters, a knot of protesters took advantage of the heavy news media presence to denounce Trump and the Republican Party with chants and signs that said "the GOP, Party of Trump."
A Ryan endorsement would help Trump and the party move past an increasingly awkward phase during which Republican officeholders and congressional candidates have publicly struggled with the decision of whether to get behind the New York real estate developer.
Trump's campaign, however, has suggested Ryan's support is not essential, pointing to the more than 10 million votes Trump has received in party nominating contests.
Full support by leading party figures such as Ryan would also help Trump build the kind of campaign infrastructure and fundraising operation he may need to compete against the likely Democratic nominee, Hillary Clinton, 68, in the Nov. 8 election.
The Hill newspaper on Thursday published a story based on an internal memo by the Trump campaign that advocated the benefits of a unified party, arguing that it would put Republicans in a better position to defeat Democrats "at every level" in the election.
For his part, Ryan must weigh the damage that endorsing Trump might do to his standing as his party's leading light on conservative policy, an image he has carefully cultivated for years. And, he may harbor hopes of running for president in 2020.
Trump last week became the presumptive nominee last week after his remaining rivals, U.S. Senator Ted Cruz of Texas and Ohio Governor John Kasich, dropped out.
Ryan was the running mate with unsuccessful 2012 Republican presidential nominee Mitt Romney, a harsh Trump critic.
A chief concern among congressional Republicans is whether Trump will be a strong enough candidate in the November election to ensure that the party maintains control of both the Senate and the House of Representatives.
While a number of elected Republicans have said they would not be willing to serve as Trump's running mate, Newt Gingrich, the former House speaker and 2012 presidential hopeful, did not rule it out. "I would certainly talk about it," Gingrich told Fox News on Wednesday.
Bob Corker, a Republican senator from Tennessee who has been mentioned as a possible Trump vice presidential choice, told reporters he would back the billionaire, but added, "I have no reason to believe that I've been considered for vice president."
House Republican Conference Chair Cathy McMorris Rodgers, who was in one of the meetings with Trump, said it was her first chance to impress upon him "the importance of championing a core value of the Republican Party: dreaming big for everyone and turning its back on no one."
(By Steve Holland and Susan Cornwell; Additional reporting by Richard Cowan, Emily Stephenson, Doina Chiacu, Susan Cornwell, Patrica Zengerle; Writing by James Oliphant; Editing by Will Dunham and Jonathan Oatis)
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