And while Trump eventually picked Pence, the vice president's modest wealth - his income reportedly has not been more than $200,000 a year for the past decade - allegedly irks Trump.
A former campaign aide said that his wife Karen was "disgusted" by the leaked tape.
The RNC began looking into replacing Trump, and top Republicans were talking about placing Pence at the top of the ticket with Condoleeza Rice as his running mate, The Atlantic reported.
Mr Pence's wife Karen was even more disgusted and called Mr Trump "reprehensible - just totally vile", the USA magazine the Atlantic reported. A White House advisor said that Trump is prone to teasing Pence about his religious beliefs and other issues.
According to the same report, published in the Atlantic on Tuesday, Trump in turn has passed judgment about the values of his vice-president.
President Trump has recently reopened the debate over the "Access Hollywood" tape, which was filmed in 2005 and shows Trump discussing groping women.
Although the rumor doesn't seem too far-fetched, Pence's press secretary Alyssa Farah emphatically denies that a coup was ever in the works. But shortly after he won the election, he began suggesting that it might not have been his voice on the tape, according to The New York Times.