One person described the session as “warm and friendly,” while the other called it a “getting to know you thing, a chance for both of them to connect.” They both noted that the presence of Karen Pence is probably a sign that the Pence family is comfortable with the prospect of the Republican governor joining the ticket, although they said they have not spoken with her.

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Both people spoke on the condition of anonymity to discuss their knowledge of the meeting, the location of which had been closely guarded for days.

Pence’s stock has been rising in Trump’s orbit, they said, describing him as respected by the candidate, despite Pence’s endorsement of Sen. Ted Cruz (Tex.) in the Republican primary.

The governor, who is in the middle of a reelection campaign, is increasingly seen as a potential smart choice by several of Trump’s aides, who have told associates that Pence’s deep ties with conservatives and his House experience would be an asset.

Still, while Pence, 57, is being vetted for the vice-presidential position, Trump’s advisers have said that he remains undecided about whom he will tap to run with him. Others being vetted by the campaign include former House speaker Newt Gingrich and New Jersey Gov. Chris Christie.

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Campaign chairman Paul Manafort, a veteran operative who is close to party leaders, is taking the lead on assisting Trump with the selection process, following Trump’s firing of Corey Lewandowski, his former campaign manager.

Compared with Pence, who has only been in sporadic contact with Trump this year, Gingrich and Christie have cultivated close relationships, with Gingrich acting as a television surrogate and occasional confidant and Christie a key player in managing Trump’s transition team, which would help to organize the federal government should Trump win the election.

In Pence’s favor is that, as a low-key Midwesterner and seasoned Republican, he would give Trump someone with governing experience and a running mate who presented voters with a temperamental and ideological contrast. A former talk-radio host and evangelical Christian who reveres Ronald Reagan, Pence has long couched his politics in the cadence of movement conservatism. Over the past decade, he has twice considered running for the White House before ultimately deciding against it.

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Working against Pence are questions in national political circles about whether he could successfully navigate the barrage of media attention and scrutiny that would come with being at Trump’s side, and whether he could successfully communicate his strident social conservatism to a fast-changing country.

In 2015, Pence was at the center of a controversial debate over an Indiana religious-liberties law seen as anti-gay that prompted protests and calls for corporate boycotts. Pence insisted that it was never the law’s intent to allow discrimination — “I abhor discrimination,” he said repeatedly at a news conference — but he acknowledged that negative perceptions had taken a toll on Indiana’s reputation.

Pence’s political network has some overlaps with Trump. Kellyanne Conway, a longtime pollster for Pence, joined the Trump campaign this past week as a senior adviser. Conway is also close to Gingrich, having worked on his 2012 presidential bid.

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The Pence campaign was unavailable for comment, but spokesman Marc Lotter told news outlets Friday that Pence would meet with Trump over the weekend and called the meeting “an invitation to spend a little time with Mr. Trump.”

Trump spokeswoman Hope Hicks declined to confirm or discuss the meeting.