The Northern Irishman is a free agent after leaving Hamilton.
Germany's Thomas Muller in action with Northern Ireland's Michael McGovern
The Sun report in their print edition, as reported by the BBC, that the former Hamilton goalkeeper has turned down a move to the Premier League club.
Northern Ireland's Michael McGovern
They claim that the free agent is not interested in joining Saints, as he does not wish to be second-choice goalkeeper to England international Fraser Forster.
Bristol City and Wigan Athletic are now in the hunt to sign the 31-year-old stopper.
McGovern is coming off the back of an exceptional Euro 2016 with Michael O’Neill’s men. His performance against Germany in the Group Stages earned him considerable praise and Southampton are believed to be one of many teams to be in the running for the player.
Southampton are looking for a new back-up goalkeeper after Dutchman Maarten Stekelenburg departed the club.
Southampton's Paulo Gazzaniga
The 33-year-old spent last season on loan at Southampton from Fulham and has now joined former Saints boss Ronald Koeman at new club Everton.
Southampton also lost regular back-up Kelvin Davis to retirement – leaving just Forster and Argentine Paolo Gazzaniga as recognised goalkeepers at the club.