Some of his most prominent supporters, including former rival Gov. Chris Christie and former Gov. Sarah Palin, would both fit that bill.
Ben Carson has also been one of Trump's most vocal supporters since he endorsed the real estate mogul after dropping out of the race himself. But Carson has never held elected office, which would appear to disqualify him for a spot on Trump's ticket.
Ohio Gov. John Kasich is still in the running for the Republican nomination himself, but has also dismissed earlier questions about the prospect of his settling for a vice presidential spot. Sen. Ted Cruz, Sen. Marco Rubio, Gov. Scott Walker and former Gov. Jeb Bush have also said in some form or another that they would not be interested in a vice presidential bid.
Trump detailed his vision of a vice president as someone with an extensive understanding of the inner-workings of Washington.
"I would like to have somebody who would truly be good with respect to dealing with the Senate, dealing with Congress, getting legislation passed, working toward something where we're not signing executive orders every three days like President Obama does," Trump said this morning.
For that, he could consider someone right from the inside, like Sen. Jeff Sessions, R-Ala., who has been a Trump supporter and whom Trump name-checked during his Indiana victory speech Tuesday night.
Or, another name that is regularly floated is that of former House speaker and former presidential candidate Newt Gingrich, who represented Georgia.
Gingrich, who lost the 2012 nomination to Mitt Romney, has praised Trump during the primary and Tuesday night told Fox News that the real estate mogul "may turn out to be the most effective, anti-left leader in our lifetime."