Banker lines up Mosley’s job – report
"Sir Fred has been sounded out over a role (in F1), possibly in a post-Max Mosley world," The Times newspaper said.
Mosley, the current president of the sport's governing body the FIA, is yet to decide whether to stand for re-election later this year.
Scotsman Goodwin, 50, was until recently the chief executive of the Royal Bank of Scotland, but is now being blamed for the institution's spectacular slump amid the global financial crisis.
But The Times said Goodwin, knighted in 2004 for services to banking, is a "petrolhead" and regular visitor to grands prix who masterminded the bank's current sponsorship of the British team Williams.
RBS's sporting ambassador, meanwhile, is the former triple world champion Sir Jackie Stewart, who has stridently called for Mosley to step down in deference to a "captain of industry, a CEO of standing" from outside F1.