Iowa Sen. Ernst told Politico: "Iowa is where my heart is. And I'm just getting started here."
Tennessee Sen. Corker told The Washington Post, "There are people far more suited for being a candidate for vice president, and I think I'm far more suited for other types of things."
Despite being denied twice in a day, it appears Trump's feelings aren't hurt too badly. He told Fox News: "I'm looking at 10 people. Three or four called me up, all want to be considered."
See some of the people still being considered:
Donald Trump's potential running mates, VPs
Donald Trump's potential running mates, VPs
Former Speaker of the House Newt Gingrich could provide Trump with exactly what he is looking for in a running mate — an experienced lawmaker who pushed legislation through Congress for years.
Though he has been actively aboard the Kasich bandwagon in recent days, Gingrich has come to Trump's defense regarding both the establishment backlash to his candidacy and the controversy the frontrunner found himself in after initially failing in a CNN interview to disavow support from former Ku Klux Klan leader David Duke.
(Joe Burbank/Orlando Sentinel/TNS via Getty Images)
Indiana Gov. Mike Pence
Pence is rumored to be one of the final few people on Donald Trump's short list to be running mate. He appeared with him mere days before Trump was expected to announce his decision, and even met with Trump's family.
Pence found himself in the spotlight in recent months after defending Indiana's religious liberty law that was criticized by many as being discriminatory against the LGBT community.
(Photo by REUTERS/John Sommers II)
A wildcard choice for sure, some began to wonder if Donald Trump might consider naming his daughter as his running mate after Sen. Bob Corker suggested the move shortly after taking himself out of the mix.
Ivanka, who would turn 35 mere days before the election, has not addressed the rumors, but brother Eric backed her.
(Photo by REUTERS/Carlo Allegri)
The 57-year-old retired lieutenant general has been advising the campaign on foreign affairs for months, but as Flynn's under-the-radar candidacy gained steam as Trump's decision drew near.
Conservative supporters have warned that Flynn isn't sufficiently tough on social issues.
(Photo by REUTERS/Gary Cameron/File photo TPX IMAGES OF THE DAY)
Gov. Chris Christie of New Jersey is the only 2016 GOP presidential candidate who has endorsed Trump since leaving the race.
Christie could help Trump with more moderate GOP voters, and he certainly has the bombastic personality that would serve as a useful surrogate for Trump, though the two also fiercely criticized each other when they were both candidates in the race.
Back in November, Trump said Christie could have a "place" on his ticket.
(Photo by John Moore/Getty Images)
Sen. Jeff Sessions of Alabama is the only sitting senator to endorse Trump — and he has already been tapped to lead Trump's national-security advisory committee.
"A movement is afoot that must not fade away," Sessions said during the Alabama rally where he announced his support last month.
Sessions is one of the staunchest supporters of Trump's hard-line plan to crack down on illegal immigration. The senator could also give Trump credibility in the South.
(Photo By Bill Clark/CQ Roll Call)
Former Sen. Scott Brown of Massachusetts was the first current or former senator to endorse Trump. He was known in the Senate as a moderate, and he could help pick up votes with some in the less conservative wing of the Republican Party.
He has supported abortion rights and is in favor of banning assault weapons, but he carries a blue-collar, populist persona. Brown memorably drove a pickup truck to campaign events during his 2010 Senate run in Massachusetts, which was to fill a vacant seat.
During a January event in New Hampshire, Trump said Brown was cut out of "central casting" and could be his vice president. Brown said at the time that Trump was "the next president of the United States."
(Photo by Keith Bedford/The Boston Globe via Getty Images)
"I was Donald Trump before Donald Trump became popular," Gov. Paul LePage of Maine said while announcing his support for the GOP frontrunner last month on "The Howie Carr Show."
The governor is comparable to Trump when it comes to provocative remarks. In January, LePage found himself at the center of a national firestorm after he made some racially tinged comments about out-of-state drug dealers who come into Maine and "impregnate a young white girl" before leaving.
"Now I get to defend all the good stuff he says," LePage has said of Trump.
LePage also entered politics after a successful business career, but he was reportedly staunchly opposed to Trump's candidacy before suddenly coming on board.
(Photo by Shawn Patrick Ouellette/Portland Press Herald via Getty Images)
Last week, BuzzFeed reported that advisers close to Huckabee thought the vice-president nod was in the cards for their guy.
Of all the former 2016 White House contenders, Huckabee may be closest to Trump ideologically. Huckabee struck a populist tone on cultural issues and, like Trump, vowed to protect Social Security and Medicare if elected.
(Photo by Christopher Furlong/Getty Images)
Aside from a few brushups in the fall, Gov. John Kasich of Ohio has barely touched Trump along the trail. The same can be said for Trump, whose most brutal attack against Kasich is that he "got lucky" because of the natural-gas reserves in his state.
It has been rumored that Trump would be interested in Kasich as his running mate, though Trump has also recently started criticizing Kasich on the campaign trail.
Kasich has the political experience that Trump says he's seeking. Kasich also hails from the Midwest, one of the most competitive regions in the past few presidential races.
(Photo by Scott Olson/Getty Images)
It has been an ongoing rumor that Gov. Rick Scott of Florida will endorse Trump after Scott wrote a gushing op-ed article in USA Today in January.
Like Trump, Scott rose to power from the business world. But Scott also has clout in the largest general-election swing state. In addition, he has six years of government experience behind him after being elected to office in 2010.
John McCain's running mate in 2008, Sarah Palin was a big get for Trump when she endorsed the frontrunner over Ted Cruz, whom she had vigorously campaigned for during his Senate run in 2012.
If Trump is interested in a sharp break with the Republican establishment, picking Palin would certainly send that signal.
It's an open question, however, as to whether she boosted or hindered McCain's run during the 2008 race.
(Photo by Patrick T. Fallon/Bloomberg via Getty Images)
Oklahoma Republican Governor Mary Fallin makes remarks before the opening of the National Governors Association Winter Meeting in Washington, in this February 22, 2014 file photo. REUTERS/Mike Theiler/Files
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One of the 10 likely being considered? Newt Gingrich. The former House speaker was expected to join Trump at an Ohio rally Wednesday evening.
And Gingrich certainly has the political background that Trump has repeatedly indicated he's looking for.
Despite that pro, there is at least one obvious Gingrich-related con. He's been a loud critic of the businessman before. Gingrich recently denounced Trump's remarks about a federal judge presiding over his Trump University cases.
Regardless, Trump is running out of time to circle one name on that VP list. He's expected to announce a running mate before the Republican Convention, which kicks off July 18.