The Euro 2016 winner recently handed in a transfer request after he claimed on social media that he hadn't been offered a new deal with the Saints with 18 months left to run on his contract.
Sam Allardyce is thought to be a big admirer of the Portugal international and wants to bring Fonte to Selhurst Park this month in order to sure up a leaky defence.
West Ham are in a similar situation to Crystal Palace, while Everton showed an interest in the summer, but are more reluctant to add a 33-year-old defender to their squad, with Phil Jagielka and Ashley Williams also over 30 years old.
Fonte was also linked with a move to Manchester United in the summer, but it seems that Mourinho has turned his attention elsewhere, probably due to the fact that Fonte is in his advancing years.
The 33-year-old has been left out of Southampton's squad for their EFL Cup semi-final first legh with Liverpool and is expected to leave the Saints this month.