Donald Trump, after effectively locking up the Republican nomination, is shifting gears to launch a national search for a running mate while teasing the possibility he could roll out key Cabinet picks as well before the July convention.

“I like the idea of doing some of this before we go into Cleveland,” Trump told Fox News’ Bill O'Reilly Wednesday night, when asked about announcing Cabinet posts early. “I could do that and I think it would be well received.”

Considering how packed the GOP field was this cycle – at one point, containing 17 candidates – Trump’s VP “short-list” is bound to be rather lengthy, especially as the campaign says they plan to consider most of his former presidential primary rivals.

Some analysts and observers put Ohio Gov. John Kasich in the top tier of that list, considering he leads the critical general election swing state. Trump said he hasn’t spoken with Kasich since he dropped out of the race Wednesday, but “would certainly consider him.”

One open question is whether many of these potential running mates would, in turn, consider Trump. Personal and partisan tensions flared throughout the 2016 primary, as barbed insults flew between Trump and whichever candidate represented the biggest threat of the moment – and his opponents frequently challenged his conservative credentials.

Several of those vanquished foes have since reconciled with Trump, including New Jersey Gov. Chris Christie and retired neurosurgeon Ben Carson, now involved in the running mate selection process.

Kasich, Christie and Carson, as well as Florida Sen. Marco Rubio and others, are among the names being floated now for some role in a hypothetical Trump administration. South Carolina Gov. Nikki Haley and New Mexico Gov. Susana Martinez are two other figures often mentioned as potential running mate picks.

However, Martinez spokesman Mike Lonergan told Fox News the governor “has said repeatedly over the years that she isn’t interested in serving as Vice President” and is “fully committed to serving the people of our state.” Haley, too, reportedly is saying she’s “not interested.” And Trump, speaking Thursday with Fox News’ Bret Baier, also said Haley is not under consideration.

Sources with knowledge of the nascent Trump VP list also disputed a CNN report that claimed Martinez, Haley and Ohio Sen. Rob Portman were early favorites.

Another recurring name in the veepstakes is former House speaker and 2012 presidential candidate Newt Gingrich.

Gingrich told Fox News on Thursday that Trump “has not talked to me about it,” noting the presumptive nominee has a “wide range” of choices. As for whom he would suggest, Gingrich said individuals like Haley, Martinez, Kasich or Wisconsin Gov. Scott Walker, among others.

He said the criteria would have to be: Can the person actually be president, can the person work with Trump and does the person help him politically?

As for the timing of any announcements about a proposed Trump Cabinet, Gingrich said making some picks early would be an “exciting” decision, adding that former New York Mayor Rudy Giuliani and Christie would be “fabulous” picks.

The New York Daily News, quoting an anonymous source, reported that Trump already has arranged to name Giuliani as secretary of Homeland Security. But a Giuliani spokesman denied that they had any meeting to discuss the post.

Asked about that possibility on Fox News – as well as the possibility of naming Christie as attorney general and Carson as health secretary – Trump called them “three very wise choices.”

But he said he has not, at this point, made that decision.

On a separate track, Trump also announced Thursday that Steven Mnuchin – CEO of investment firm Dune Capital Management and a former partner at Goldman Sachs – will serve as his campaign’s national finance chairman.

Fox News’ Faith Mangan and Serafin Gomez contributed to this report.