The Saints are looking to secure a Europa League place this season, and having seen their form tail off slightly in the second half of the season, manager Ronald Koeman will be looking to improve his squad this summer.
The future of centre back Toby Alderweireld will be a big factor in the Saints' summer spending, with the Atletico Madrid-owned defender currently on loan at St. Mary's Stadium.
With recent reports from The Telegraph suggesting that Tottenham are interested in signing the Belgian this summer, Southampton may need to start looking at alternatives in case he doesn't stay on the South Coast.
Now, the Mirror believe that Koeman's men have identified Basel star Schar as a possible replacement for Alderweireld after another strong season.
The Swiss international, 23, has just one year remaining on his contract with Basel, but could leave this summer, with plenty of interest likely - especially considering that he has five goals in 12 games for Switzerland despite being a centre back.
The Sun recently suggested that West Ham were leading the charge for Schar, but the Mirror believe that the uncertainty surrounding Sam Allardyce's future as Hammers boss has allowed Southampton to move in and line up a deal for Schar this summer.
Until West Ham figure out who will be managing them next summer, they may find it difficult to secure new signings - and that may be the case with Schar as Southampton join the race for his signature.