Romney, a former Massachusetts governor, is credited with firing the opening salvo when he questioned the pro-life credentials of former Thompson, a former U.S. senator from Tennessee, The Los Angeles Times reported Friday.
Thompson, who is expected to enter the GOP race, first denied and later confirmed that he had lobbied for the National Family Planning and Reproductive Health Association in the early 1990s.
He would be the only Southerner in the race, if he enters, and polls show he has a strong appeal to religious conservatives.
Many on the right expressed unhappiness with the party's front-runner, former New York Mayor Rudy Giuliani, an abortion rights supporter.
"In many ways, the Romney campaign and the emerging Thompson campaign are on a collision course when it comes to campaigning for this constituency of conservative Christians," John C. Green of the Pew Forum on Religion & Public Life, told the Times.