It's not clear whether the freshman Republican senator and tea party star has received paperwork from the campaign or is just being scrutinized through the wealth of publicly accessible information about her.
New Jersey Gov. Chris Christie, former House Speaker Newt Gingrich and Indiana Gov. Mike Pence are also being considered, the source said. A senior Trump campaign adviser confirmed Pence is being formally vetted.
A senior Trump adviser confirmed that Ernst is being considered, but that it's not clear that Ernst is interested "as opposed to others who have said, 'Thanks, let's play ball.' "
Ernst's office has not responded to requests for comment.
Alabama Sen. Jeff Sessions, who was the first U.S. senator to endorse Trump and who has often been floated as a potential running mate for the billionaire, could be a fifth candidate but his name wasn't listed among those who had received the vetting paperwork, the source said.
The source added: "Pence is the obvious leading candidate for those who want a safe choice and someone who will appeal and pass the test of GOP establishment."
There are people close to Trump who are pushing for Pence, the source said.
Christie is more likely than Gingrich to be the second choice because "Newt is Newt," meaning he'll speak his mind, and will not be as loyal to Trump, the source added.
Gingrich, for instance, told an audience at the Aspen Ideas Festival: "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.' "
One commonality between all five potential choices is their political experience, a trait Trump has often said is a key factor in his decision as he looks ahead to advancing his legislative agenda through Congress.
"Only one person, Trump, is making the decision and he can change his mind from minute-to-minute," the source said.
Trump met with Pence Saturday at the Trump National Golf Club in Bedminster, New Jersey, a source with knowledge of the meeting told CNN.
The meeting went well, according to two sources familiar with it.
Two sources said Pence was invited to stay for golf.
"On a scale of 1-10, it was a 9," one source said.
"The Pences enjoyed spending warm, productive time with the Trumps," said Pence spokesman Marc Lotter in a statement on Sunday. "They talked about policies that are working in Indiana and the future of this country. Nothing was offered and nothing was accepted."
Under an Indiana law that prohibits candidates from running simultaneously for state and federal offices, Pence faces a July 15 deadline to withdraw his name from his re-election race for governor if he's to join Trump's ticket — at least in Indiana, a reliably Republican state.
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.