Some possible picks have indicated openness to the job; other prominent Republicans have forcefully said they would not join Trump on a ticket.
Trump plans to announce his No. 2 at the Republican National Convention in July and has asked his campaign manager, Corey Lewandowski, to head the team vetting candidates.
Several names have been floated, some by Trump himself. What are their thoughts? Here's a look.
Open to the idea
Former Arizona Gov. Jan Brewer
Jan Brewer, whose sharp attacks on illegal immigration resemble Trump's, has lauded his sometimes caustic rhetoric.
In 2010, she signed a law that made it a crime for immigrants in the country illegally to seek work or travel without carrying immigration papers. The law also required police to determine the immigration status of someone arrested or detained.
Opponents of the law said it would lead to more racial profiling by police. The U.S. Supreme Court struck down much of the law in 2012 although it upheld the part allowing police to check people's status.
Picking Brewer would emphasize what has already been a key issue for Trump.
In an interview with Fox News in which he was asked whether any women were on his list of vice presidential possibilities, Trump referred to Brewer as "fantastic."
She has indicated interest. "I would be willing to serve in any capacity that I could be of help," she told CNN recently.
New Jersey Gov. Chris Christie
Shortly after suspending his own presidential campaign, Chris Christie became the first prominent figure in the Republican establishment to endorse Trump. He has traveled to primary states to campaign on Trump's behalf and stood at his side during news conferences.
Christie, whose blunt style is similar to Trump's, currently spearheads the wealthy businessman's transition team.
Christie's state -- a Democratic stronghold -- doesn't seem like a likely target for Trump in a general election, and Christie is currently extremely unpopular at home. His brash manner could be an asset to Trump's campaign, however, particularly in one of the vice presidential nominee's traditional jobs -- attacking the other side.
"He was an early endorsement and a very enthusiastic one," Trump said of Christie in a recent interview on Fox News. "He is a friend of mine and he is a very good guy and a talented guy and he is helping us a lot."
Oklahoma Gov. Mary Fallin
Mary Fallin, who also served in Congress, is relatively unknown on the national stage and is not particularly popular in her home state. She became a subject of vice presidential chatter after former South Carolina Lt. Gov. Andre Bauer suggested her while speaking on CNN.
"Great job and advice," Trump wrote in a Twitter message after Bauer's appearance. In his Fox interview, Trump mentioned Fallin, along with former Arizona Gov. Brewer, as women he would consider.
Fallin called it an "honor" to be mentioned as a possible vice presidential pick and and has indicated an openness to serving alongside Trump.
Former House Speaker Newt Gingrich
Gingrich, whose quest for the Republican presidential nomination in 2012 faltered, has deep roots in Washington and could serve as a bridge to some establishment Republican figures, although he also would bring considerable controversy with him.
At 72, he would be older than Trump, who is 69.
Gingrich has praised Trump for running "one of the most remarkable campaigns in American history."
Alabama Sen. Jeff Sessions
Sessions is in lockstep with Trump on an issue that's become the core of his campaign: illegal immigration.
From building a wall along the U.S.-Mexico border to deporting immigrants, Sessions is a fierce advocate of Trump's immigration proposals. Sessions, who has spent nearly two decades in Washington, advises Trump on foreign policy.
He has told reporters on several occasions that he's willing to undergo the vetting process, should Trump consider him as a running mate.
On the fence
Iowa Sen. Joni Ernst
An Iraq war veteran serving her first term in the Senate, Ernst remained neutral throughout the caucus process in her home state. (She did, however, make a brief appearance at a rally for Florida Sen. Marco Rubio.)
Many establishment Republicans, including Sen. John McCain of Arizona, the 2008 GOP nominee, and Iowa Gov. Terry Branstad, have floated Ernst's name as a possible running mate for Trump. Ernst is viewed by them and others as a rising star within the Republican party and might be able to help Trump improve his dismal poll numbers with women voters.
"I'm just focusing on Iowa right now," Ernst said in a brief interview with Politico this month, declining to rule out becoming Trump's No. 2. "That is my concern."
Florida Gov. Rick Scott
During the March primary in his home state, one that pitted Trump against Florida Sen. Marco Rubio, Rick Scott remained neutral. But a day after Trump easily won Florida, Scott endorsed the billionaire mogul, who regularly stays at his Mar-a-Lago Club in South Florida.
Scott's approval ratings in Florida are far from stellar. Recent polls show him below average in popularity, although improved from last year. Still, he represents Florida, a perennial swing state with 29 electoral votes that could be pivotal in November.
Scott has said he's going to remain governor through the end of his term in 2018, but he has also told reporters he'd "do anything I can to make sure [Trump] wins.”
South Carolina Gov. Nikki Haley
Nikki Haley is a potentially transformational figure in the Republican party -- the daughter of immigrants from India who became the state's first female governor. Her endorsement was coveted by GOP presidential hopefuls ahead of the state's primary.
She backed Sen. Marco Rubio, and during her rebuttal of President Obama's State of the Union address in January, threw an indirect jab at Trump.
"During anxious times, it can be tempting to follow the siren call of the angriest voices," Haley said during that nationally televised speech.
She told local reporters this month that she has no interest in a vice presidential slot.
Ohio Gov. John Kasich
The Ohio governor has consistently blasted Trump's rhetoric toward women and immigrants, calling it divisive and un-presidential. John Kasich, who leads a critical swing state, has noted that he will support the Republican nominee this fall.
But what about serving as Trump's No. 2?
"Zero chance," he said last month in an interview with CBS.
Florida Sen. Marco Rubio
Marco Rubio suspended his campaign in March, but has remained a critic of Trump.
In recent days, Rubio repeated that he will honor his pledge to support the party's nominee, but that he has no interest in becoming Trump's running mate. He expressed reservations about the businessman's views on foreign policy.
"He will be best served by a running mate and by surrogates who fully embrace his campaign. As such, I have never sought, will not seek and do not want to be considered for vice president," he wrote in a Facebook post.