The report claims that Alcacer has been told that he can leave Barcelona in January and find a new club.
Southampton and Crystal Palace have both been put on high alert by this and are now considering a January swoop.
The striker has failed to live up to the hype since joining Barcelona for big money from Valencia last summer.
He scored eight goals for Barcelona last season but was often criticised by fans for his performances over the course of the campaign.
Southampton are now looking to make the most of the opportunity and bring him to England.
The 24-yer-old looks set to leave and Southampton have been struggling for goals.
The Saints have scored in three of their eight matches in all competitions this season – scoring four goals in total.