Viewed as a safe pair of hands, Pence has diverging views with Trump on his proposed Muslim ban and trade, and is more socially conservative, but he could help unify a divided party behind Trump's White House bid.
Trump is due to make his official announcement on his choice on Friday in Manhattan.
Trump's advisers told national party officials that he had settled on Pence, according to two Republican sources familiar with the campaign's operations.
Contradicting this, campaign spokesman Jason Miller tweeted, "A decision has not been made by Mr Trump. He will be making a decision in the future."
Trump, a New York businessman, is to be formally nominated as the party's candidate for the November 8 presidential election at the Republican National Convention next week in Cleveland. Traditionally, the vice presidential choice is used to build enthusiasm among party loyalists.
Trump's choice of running mate is seen as especially critical because his defeat of 16 rivals in the Republican primary race left the party divided. Some party leaders are still uneasy about some of his campaign positions and free-wheeling statements, such as his comments on Muslims and immigrants.
"Pence is Donald Trump's straight man," said Republican strategist Ron Bonjean. "He'll be able to defend him as well as be a cheerleader but do it in a calm, cool, collected manner that will preserve his credibility."
Trump (70) had whittled his short list down to three political veterans: 57-year-old Pence, 57; former House of Representatives Speaker Newt Gingrich (73); and New Jersey Governor Chris Christie (53).
Gingrich, who met Trump on Wednesday, said on a Facebook Live session that he had yet to hear from Trump. He said he had told Trump that he needed to decide whether he wanted "two pirates" on the same ticket. Both men have been political mavericks.
Christie told MSNBC earlier he would be disappointed if not picked. "I'm not going to say it won't bother me if I'm not selected. Of course it bothers you a little bit," Christie said.
Pence, a former congressman, is seen as a safe choice, not too flashy but popular among conservatives, with Midwestern appeal and the ability to rally more party faithful behind Trump. The businessman has never held elected office.
"He's a good, safe, solid conservative," said Republican strategist Scott Reed.
Pence also could give a boost to Trump's campaign fundraising efforts as he challenges the well-organized effort of his Democratic rival, Hillary Clinton. Pence has strong ties to billionaire donors Charles and David Koch, including current and former staff members who have worked for them.
DIFFERENCES
However, Pence is to the right of Trump on social issues, having signed restrictive abortion legislation and pushed to defund the Planned Parenthood women's healthcare organisation, whose services include providing abortions. Trump has said he opposes abortion, but his views have been inconsistent, and he has said Planned Parenthood provides some valuable services.
Pence has also criticised Trump's proposal to ban Muslims temporarily from entering the country. And in 2006, he introduced immigration legislation that would let illegal immigrants apply for US work visas if they left the country for a period, a plan that was criticized by some conservatives. This contrasts with Trump's strong stance on immigration, marked by his pledge to build a wall along the border with Mexico.
Pence has also voiced support for free trade deals like the Trans Pacific Partnership. Trump supports free trade but says he wants to renegotiate trade deals to make them more favorable to the United States.
As is traditional in the vice presidential auditioning process, Trump held a flurry of meetings in recent days with potential candidates. Pence and Trump met on Wednesday at the governor's residence in Indianapolis, joined by members of Trump's family.
Pence had backed a Trump rival, Senator Ted Cruz of Texas, in April before the Indiana primary, but praised Trump and said he would work on behalf of the eventual Republican nominee. Trump won Indiana anyway, prompting Cruz to drop out of the race to be the party's nominee.
Pence had considered running for president himself in 2016 before deciding to run for re-election as governor. Conservatives had urged him to seek the White House, but missteps last year related to an Indiana law seen as anti-gay hurt his national profile.
This year, he was the target of a mocking social media campaign by women outraged at a law he signed creating new restrictions on abortions. Feeling that the law invaded their privacy, women responded by calling Pence's office to describe their menstrual periods or tweeting similar messages.
Pence ran unsuccessfully for Congress twice before he won election to the House of Representatives in 2000, where he was chairman of the Republican Study Committee, a group of conservatives.
