The nomination is expected to be formalized before Trump meets with Pope Francis on May 24 in Rome, and once the Office of Government Ethics gives its approval, CNN reported Sunday.
A self-described devout Catholic, Gingrich, 51, currently heads Gingrich Productions, a multimedia company.
Newt Gingrich, 73, was one of the few big-name Republicans to support Trump during the 2016 presidential campaign.
He was speaker of the House of Representatives -- second in line in the US presidential succession chart -- from 1995 to 1999, and was notably vociferous in his calls to impeach president Bill Clinton over the Monica Lewinsky sex scandal.
The potential nomination carries some controversy: Callista Gingrich is the former speaker's third wife, and had a multiyear sexual relationship with him when she was a young congressional aide named Callista Bisek and he was still married to his second wife.
After a messy divorce from his second wife in 1999, he married Callista the following year. He credits Callista with his conversion to Catholicism.
Trump mentioned the possibility of the ambassadorship in January, but the Gingrichs have been so frustrated with the slow vetting process that they have threatened to withdraw Callista's name, The New York Times reported.
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