Saints and Spurs are close to agreeing a £12million fee for the defender , with Pierre-Emile Hojbjerg set to move the other way in a separate deal worth around £15m.
However, Fulham are considering entering the bidding for Walker-Peters following their promotion to the Premier League via a 2-1 win over rivals Brentford in the Championship play-off final.
Fulham missed out to Saints on a loan deal for Walker-Peters in January but the Cottagers remain in the market for a right-back, with manager Scott Parker and his assistant Matt Wells both admirers of the 23-year-old, having worked with him in Spurs' academy.
Negotiations for Hojbjerg and Walker-Peters have been closely linked but the deals have deliberately been kept separate to ensure that one is not dependent on the other.
Spurs are confident that they will land Hojbjerg, who has one year remaining on his contract and wants to move to north London, regardless of whether Walker-Peters ends up at St. Mary's.
The former England U-21 international has enjoyed his time on the south coast but could be attracted by the chance to stay in London and be reunited with Parker and Wells.
Both clubs have also expressed an interest in signing Spurs youngsters Ryan Sessegnon, Jack Clarke and Oliver Skipp on loan next season, although the latter two are more likely to join Championship clubs.
Mourinho is willing to give teenager Clarke - who is also attracting interest from West Brom - a chance to win a place in his Spurs squad after an unhappy season on loan at former club Leeds and then QPR, making just 10 appearances in total.
The manager wants more competition in his front four and is therefore set to run the rule over Clarke in pre-season before deciding if he will spend another year out on loan.