By Jeremy Diamond and Will Cadigan CNN

WASHINGTON (CNN) -- Donald Trump says he has narrowed down his list of potential running mates to "five or six" names and his campaign is beginning to vet those candidates.

So who's on the shortlist?

The presumptive GOP nominee isn't revealing any names, but he and his top campaign aides in recent days have offered some hints.

Trump, who has made his business resume and outsider status a cornerstone of his campaign, has repeatedly emphasized that he wants a running mate with political experience and longstanding relationships on Capitol Hill who can help him pass his ambitious agenda through Congress. He has also said politicians who have long been in the limelight would be easier to vet.

Trump campaign manager Corey Lewandowski, who has been tasked with overseeing the vice presidential vetting process, said Tuesday on the Fox Business Network that all of the names on Trump's shortlist "have had elected office experience at one level or another."

Lewandowski also said the campaign wasn't looking to check any boxes: "It's not about gender, not about geographic representation -- Who best is a partner to bring our country back from where it is."

Trump campaign sources have also told CNN that Trump would lean toward picking a running mate with whom he gets along and trusts -- not just a candidate who brings the best political benefit in November or the most experienced resume in January.

Here's our take on the "Trump 6" -- in no particular order -- based on that trail of breadcrumbs.

New Jersey Gov. Chris Christie

Christie was the first of Trump's primary rivals to endorse him and immediately became Trump's most prominent backer. The New Jersey governor, also known for his brash, straight-talking style, would help Trump preserve his outsider image while also bringing the Washington relationships and political experience Trump is looking for.

Beyond the connections he brings from his time as chairman of the Republican Governors Association, Christie played up his national security experience from dealing with terrorism cases as a U.S. attorney during his 2016 bid.

Trump has repeatedly been asked about Christie -- and says he hasn't ruled him out from the veepstakes. Another encouraging sign for Christie: he's already been entrusted by Trump to lead his transition team, should he win the White House.

Oklahoma Gov. Mary Fallin

Fallin fits the bill for a lot of reasons: she's a governor with decades of political experience that includes two terms in the House of Representatives and one term as chairwoman of the National Governors Association. Plus, she's a woman, which could help Trump, who currently has a dismal approval rating among women, as he faces a likely general election against Hillary Clinton.

Fallin also seems pretty open to the idea. She's said she's "behind Donald Trump 100%" and "would be very honored if I were to receive a call saying I need you to help make America great again."

Trump, for his part, tweeted last month that the idea of picking Fallin as his running mate was "great advice."

Sen. Bob Corker of Tennessee

Corker shocked many in the political world when he offered quick praise of Trump's foreign policy speech in Washington late last month. The Senate Foreign Relations Committee chairman has since been advising Trump on foreign policy and would bring the legislative chops -- including a willingness to work across the aisle -- Trump says he's looking to add to his ticket.

Corker's spokeswoman said the senator "has no reason to believe he is being considered for vice president."

Former House Speaker Newt Gingrich

Gingrich is another choice that would give Trump the legislative experience he seeks while still maintaining his outsider image. Gingrich has been around the block in Washington but is still seen as an insurgent force.

He's said he "isn't interested," but would "obviously have to listen carefully if he (Trump) called."

Still, at 72, Gingrich's age could be a factor for Trump, who is 69.

Sen. Jeff Sessions of Alabama

Sessions was the first senator to endorse Trump, doing so in February before the crucial Super Tuesday primary. And the two have enjoyed a close relationship prior to that, beginning when Sessions helped Trump craft his immigration policy last summer. Sessions has since taken the helm of Trump's team of foreign policy advisers and would bring decades of legislative experience to the ticket.

Sessions has suggested he's unlikely to be picked, but has said it would be "an honor."

At 69, Sessions and Trump are the same age.

Sen. Joni Ernst of Iowa

While she doesn't bring decades of national political experience, the freshman senator would boost Trump's lack of national security experience with her military background.

Ernst has at times been critical of Trump, encouraging women "to stand up and say, 'You know what, I'm not going to put up with his nonsense' " after an April poll showed 70% of women had a negative view of Trump. As a past critic, Ernst would be uniquely positioned to help him with the key demographic. Plus, she's from the potential battleground state of Iowa.

Trump hasn't floated her name, but Ernst has left the door open when asked about the possibility.

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