The 26-year-old is expected to leave the Saints in the current market following their relegation to the Championship, and his future has been the subject of much speculation.
According to The Sun, Fulham are currently leading the race for the Englishman, with Southampton believed to be willing to sell for a fee in the region of £20m.
The report claims that Fulham are hoping of agreeing a fee of between £15m and £18m for the former Tottenham Hotspur youngster.
Walker-Peters - also believed to be of interest to Lens and Roma - has made 110 appearances for the Saints in all competitions, including 38 outings during the 2022-23 campaign.
Elsewhere, Aston Villa are reportedly interested in signing Tottenham attacker Giovani Lo Celso during this summer's transfer window.
The 27-year-old is back at Spurs this summer following a loan spell at Villarreal, but he is not thought to be in Tottenham's plans for the upcoming season.
According to Estadio Deportivo, Villa are 'very interested' in the Argentina international, with head coach Unai Emery viewing him as the ideal signing during the summer market.
Lo Celso has featured on 84 occasions for Tottenham since his arrival from Real Betis, while he made 51 appearances for Villarreal during a loan spell.
Staying with Spurs, Davinson Sanchez is reportedly heading towards the exit door this summer.
The 27-year-old has been with the North London club since 2017, featuring on 205 occasions in all competitions.
There is currently speculation surrounding the Colombia international's future, though, as he was restricted to 24 appearances last term, with only 18 of those coming in the Premier League.
According to the Daily Mail, Spurs have now accepted a £12.9m offer from Spartak Moscow for the centre-back, but it remains to be seen whether he wants to move to Russia.
The report claims that Galatasaray are also showing an interest in the defender, who has seemingly played his last game for Tottenham.
Meanwhile, Wolverhampton Wanderers are reportedly preparing to sell Daniel Podence during this summer's transfer window.
The attacker has not travelled on Wolves' pre-season trip to the Algarve amid the speculation surrounding his future.
According to the Daily Mail, Julen Lopetegui's side are preparing to sell Podence in the coming weeks.
The report claims that the 27-year-old, who is out of contract next summer, is not in Lopetegui's plans for the 2023-24 campaign.
Podence arrived at Molineux from Olympiacos in 2020, and he has represented the club on 105 occasions in all competitions, including 37 appearances during the 2022-23 campaign.