Telegraph journalist Matt Law has claimed that the Blues 'have made enquires' about the former Tottenham defender. However, they have made similar advances for numerous players all summer.
Law also claimed that the Saints' valuation of the 25-year-old is high, as he still has three years left on his current deal. Walker-Peters was one of the leading lights at St. Mary's last season, as he made 37 appearances across the campaign.
The former Tottenham defender also made his senior England debut earlier this year. The full-back was also part of the Spurs squad that reached the 2019 Champions League final under Mauricio Pochettino. He then moved to the South Coast in January 2020.
Walker-Peters would be an extremely valuable addition to Thomas Tuchel's side, with Chelsea in desperate need of defenders during the current window. Antonio Rudiger and Andreas Christensen have already departed. SPORT have also reported that full-backs Cesar Azpilicueta and Marcos Alonso could join Barcelona.
Thomas Tuchel demands more signings from Chelsea board ahead of new season
The Blues are coming off the back of a turbulent campaign that was severely affected by previous owner Roman Abramovich.
However, despite the endless distractions off the pitch, Tuchel still managed to lead his side to a third-placed finish while also reaching both domestic cup finals. Pre-season, though, has been a struggle for the Blues, whose struggles were compounded by a brutal 4-0 defeat to Arsenal on their US tour.
Following the loss, Tuchel set off on a spectacular rant regarding his team's lack of summer business. The German boss said at a press conference (as per The Guardian):
“I am far from relaxed. We were simply not good enough. We were simply not competitive. I look at the last season and parts of the game where we struggled, parts of the season where we struggled, and then we got sanctioned and players left us and some players are trying to leave us, and this is where it is."
"So we had an urgent appeal for quality players, a huge amount of quality players. We got two quality players but we are not competitive like this. Unfortunately, you could see it today.”
The Blues have so far brought in Manchester City forward Raheem Sterling as well as Napoli defender Kalidou Koulibaly.