The Saints are said to be stalling over contract negotiations with the in-form winger, risking losing him for free at the end of the season.
Both West Ham and Newcastle are thought to be interested, and could take advantage of the uncertainty.
Puncheon is believed to be keen on staying at St Mary’s Stadium, but negotiations regarding a new deal are some way off of being concluded.
The 26-year-old has been in fine form this term, notching six goals for the newly promoted side.
He has also weighed in with a number of assists, making him, arguably, one of the club’s key men under new boss Mauricio Pochettino.
Despite this, his frosty relationship with chairman Nicola Cortese, who had placed him on the transfer list, may see him shown the door.
He is said to be happy to extend the former MK Dons star’s stay on the South Coast, but does not want to do so until the end of the season.
West Ham boss Sam Allardyce is believed to be keen to draft in fresh attacking options, whilst the Magpies are also interested in adding more creativity and pace to their ranks.