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Donald Trump's VP search is well underway two weeks before his party's convention

Trump met with Indiana Gov. Mike Pence and Iowa Sen. Joni Ernst over the weekend

Washington CNN  — 

Donald Trump’s much anticipated vice presidential pick is expected to be announced next week, a Trump adviser told CNN Tuesday.

The presumptive Republican nominee will choose from a narrowing list of potential running mates that represent a wide range of styles.

Does Trump want to double the bombast by picking a brash straight-talker who mirrors his own style, like New Jersey Gov. Chris Christie or former House Speaker Newt Gingrich?

Does he prefer someone whose Washington experience and message discipline would counter his freelancing, like Indiana Gov. Mike Pence?

Or does he want to elevate a rising star, like Iowa Sen. Joni Ernst or Arkansas Sen. Tom Cotton?

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Just two weeks from the Republican National Convention, Trump’s vice presidential search – and speculation about who he’ll pick – is at a fever pitch.

Trump met in person over the holiday weekend with Pence and Ernst, and kicked up speculation when he praised Ernst and Cotton on Twitter – something he did again on Tuesday.

“It was great spending time with @joniernst yesterday. She has done a fantastic job for the people of Iowa and U.S. Will see her again!” Trump tweeted.

A source familiar with the vetting process told CNN that Republican National Committee Chairman Reince Priebus and campaign chairman Paul Manafort were in the meeting with Ernst. Another source told CNN that the meeting was in part to discuss whether Ernst was interested in being formally vetted and it’s now up to her to make a decision about that.

But sources said they weren’t clear what her decision would be.

People close to Ernst said she didn’t seem particularly interested in being vetted prior to Trump reaching out. But she may have re-evaluated in light of the meeting.

On Tuesday, another oft-mentioned potential running mate, Sen. Bob Corker of Tennessee, campaigned with Trump in North Carolina. Corker, chairman of the Senate Foreign Relations Committee, also introduced the presumptive Republican nominee at a Raleigh rally. Two sources familiar with the vetting process told CNN Wednesday that Corker was “very seriously” being vetted and has submitted a lot of information to Trump’s campaign. He’s on a very short list, the sources added.

Gingrich will campaign with Trump Wednesday in Ohio, campaign spokesman Jason Miller said.

But Trump’s campaign has been quiet about who he’ll pick – and when he’ll make an announcement.

His former campaign manager, Corey Lewandowski, who is now a CNN political contributor, told CNN’s Brianna Keilar Sunday on “State of the Union” that the key to Trump’s selection will be his comfort level.

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“It’s really going to come down to the personal relationship that he and Mr. Trump could have to find out if they could work well together,” said Lewandowski, who was fired last month.

He emphasized the importance of Trump tapping a politician with ties to Washington, who could help advance his legislative agenda in Congress.

“Having someone who has those relationships in Washington is going to be a critical component of his presidency,” he said.

Pence, Ernst and Cotton were late additions to the vice presidential speculation – with NBC News reporting late last week that Pence was under consideration and sources telling CNN that Ernst is also being considered.

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Pence and his wife Karen met with Trump at his New Jersey golf club on Saturday. The huddle came as a source told CNN some in Trump’s inner circle are pushing for Pence.

Michael Caputo, a former adviser to Trump’s campaign, echoed that sentiment in an interview with CNN’s John Berman on Monday.

“I am a Mike Pence guy,” Caputo said. “First of all, he puts away some of the problems we have with the conservatives in the party who fear that Donald Trump is too moderate on certain policy issues.”

Caputo added: “He is very likeable. People across the country involved in Republican politics who found him to be a very personable man. In addition, he has got no question as a leader of the Republican Conference in the House, before he left the United States Congress, he brings the lawmaking ability, the ability to work the House and work the Senate, that Donald Trump doesn’t bring as a man who doesn’t come from a career in politics.”

Pence was the No. 3 House Republican, leading the conference and holding particular sway with social conservatives and tea partiers, before departing Washington to run for governor of Indiana in 2012.

He wouldn’t comment on being considered as Trump’s running mate Sunday night in Indiana.

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“I love my job. I love my country. I’m not going to get into hypotheticals,” he told reporters there.

Ernst, meanwhile, came onto the political scene in the 2014 midterm elections – gaining national attention when she compared her plans for Washington to castrating pigs on Iowa farms.

She also has an advantage others on Trump’s list don’t: Location. Iowa, unlike Pence’s Indiana, Gingrich’s Georgia, Christie’s New Jersey and Cotton’s Arkansas, is a swing state.

After the meeting on Monday, Ernst said she had “a good conversation with Donald Trump.”

“We discussed what I am hearing from Iowans as I travel around the state on my 99 county tour, and the best path forward for our country,” she said in a statement. “I will continue to share my insights with Donald about the need to strengthen our economy, keep our nation safe, and ensure America is always a strong, stabilizing force around the globe.”

Trump also doled out some conspicuous praise for Pence and Cotton on Twitter Monday.

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Trump tweeted his approval of Cotton’s performance in an interview on Meet the Press on Sunday: “Senator Tom Cotton was great on Meet the Press yesterday. Despite a totally one-sided interview by Chuck Todd, the end result was solid!”

Cotton discussed on NBC his support for Trump, but when asked to explain why he should be president, the Arkansas senator said only that “Donald Trump can ultimately make the case for himself.”

Pressed on why his support seemed less than full-throated, the hawkish Arkansas freshman jokingly offered that “maybe I just don’t demonstrate enthusiasm much in life, especially in such dangerous times as these.”

Gingrich, the former House speaker, was critical of Trump – but also laid out a hypothesis for the campaign against Hillary Clinton – on stage at the Aspen Ideas Festival in Colorado on Saturday.

“Trump’s job is frankly to quit screwing up and get the election down to three or four big issues, all of which come down to a single concept: ‘enough,’” Gingrich said.

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He made the case against taking the job, too, noting that he has a “terrific life right now” and said if you’re vice president, “you actually have to work.”

“I used to be a congressman. I know what it’s like to deal with congressmen. I don’t know I want to spend the next four to eight years of my life dealing with congressmen,” he said. “I think I’d rather read books and write books and make movies.”

Trump could announce his choice of a running mate before the party’s convention to help drum up excitement heading into Cleveland, a source familiar with the process told CNN on Friday. The billionaire and his aides had previously said the pick would likely be announced at the convention, which starts July 18.

CNN’s Manu Raju contributed to this report.