Mr Thornton had been seen as one of the favorites to be named in place of outgoing chairman Stephen Green but finance director Douglas Flint was appointed to the role instead last week.
Insider sources told the Financial Times that Mr Thornton - who used to work at Goldman Sachs and has an extensive contact list in China - will step down in spring unless more importance is given to the financial institution's Chinese operations.
"He won't behave irresponsibly but he doesn't plan to stand for re-election," said one insider.
Chief executive Michael Geoghegan has also recently confirmed his departure from HSBC in the fallout from the race to succeed Mr Green, with the Guardian reporting that the former has walked away with around Â£17 million ($26.9 million) in cash and shares.
By Tony Aynsley