The newspaper claims that some national motoring bodies are considering leaving the FIA and setting up a breakaway organisation if Mosley succeeds in clinging to office.
The American Automobile Association, which has already called on Mosley to step down, warns that the ongoing controversy threatens to “significantly impact” the FIA.
“This in turn could give way to a new international umbrella organisation,” AAA spokeswoman Yolanda Cade told the Sunday Times.
“In the interests of preserving the FIA, the AAA believes the most appropriate course of action would be for Mosley to step down.”
Mosley will face a vote of confidence in his presidency by the FIAs worldwide membership on June 3 in Paris following the News of the World's allegations that he took part in a “Nazi-style orgy” in a Chelsea flat.
He denies there were any Nazi connotations to his actions and has served legal proceedings against the tabloid citing a breach of privacy.
But he has faced mounting calls for his resignation from national automobile and motorsport bodies, with the USA, Germany, Spain, South Africa, Holland, Israel, Canada, Australia and New Zealand all questioning whether he should continue in office.
However, Mosley insists he retains enough backing within the governing body to win a vote of confidence, recently claiming that he had received seven supportive letters from member clubs for every one that called on him to step down.
Only two voting organisations – Brazil and the United Arab Emirates – have publicly pledged their allegiance to Mosley, but he is believed to have a strong power base among smaller motorsport clubs that hope to attract FIA-sanctioned events in their countries.
The AAA, with 50 million members, is the FIAs largest constituent body, but carries no more voting weight than the other 221 member clubs.
However its spokeswoman claimed dissatisfaction with Mosley was not limited to the larger high-profile clubs.
“The wisdom of individual club representatives and their ability to deal with the current controversy is not a function of the size of their organisation,” said Cade.
“Large and small clubs have expressed their indignation over what has occurred.”
Formula 1 teams have no direct influence over Mosleys fate, although four of the six car manufacturers involved in the sport – Mercedes-Benz, BMW, Honda and Toyota – have expressed concern that the scandal risks damaging F1s image.
Last Saturday at the Spanish Grand Prix, representatives of 10 of the 11 teams discussed the situation with F1 impresario Bernie Ecclestone at a meeting in Toyotas motorhome.
Several media outlets reported that seven teams were willing to sign a letter calling for Mosley to stand down, but that three – Ferrari, Williams and Toro Rosso – declined to do so.
Some sources claimed that Ecclestone offered to withdraw his backing for Mosley if the teams could agree a united position, and that only the lack of unanimity prevented him from doing so.
However, Ecclestone dismissed claims that he was ready to abandon his long-time friend and business partner as “a load of nonsense”.