The Hamburg attacker is currently sidelined after he sustained a cruciate injury after celebrating a goal.
Both the Hammers and the Saints were linked with the player throughout the summer but a deal never materialised with Hamburg asking for around £8 million.
Muller, who is believed to be one of Hamburg’s top earners, is out of contract at the end of the season and the club have blocked an extension on his current deal.
The Bundesliga side are struggling financially and do not want to commit to a player when they are not sure when he will return.
With no new deal on the table, Muller will be able to discuss a summer move in January and possibly agree pre-contract terms.