Story highlights The source added that Indiana Gov. Mike Pence is very high on Trump's VP list

Earlier in the week, Gingrich told Fox News that if asked, he would serve as Trump's vice president

(CNN) Newt Gingrich is "actively lobbying" to be Donald Trump's running mate, a source familiar with the process told CNN Saturday.

The source added that Indiana Gov. Mike Pence is very high on Trump's VP list and is getting the "full vet." But Pence isn't actively lobbying for the job because it's not his style, the source added.

And retired Lt. Gen. Michael Flynn is also being considered as a Trump running mate, but the source said it is "not clear how intensely," an indication the general is on the list but may not get the final nod.

Sources have also said New Jersey Gov. Chris Christie, an active surrogate for Trump, is not yet out of the running. He will campaign with Trump in Virginia on Monday, campaign spokeswoman Hope Hicks said.

Earlier in the week, Gingrich told Fox News that if asked, he would serve as Trump's vice president

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