The Hammers have allegedly identified the 27-year-old as a leading target during the current market.
According to talkSPORT, Southampton have asked for £10m plus Flynn Downes for the Englishman, who has entered the final 12 months of his contract at St Mary's.
Russell Martin's side are said to be unwilling to run the risk of losing him on a free transfer next summer, meaning that a move during the current window is likely.
Walker-Peters was in strong form for the Saints during the 2023-24 campaign, scoring three goals and registering three assists in 47 appearances in all competitions.
In total, the defender has made 167 appearances for Southampton, featuring on 115 occasions in the Premier League during his time at St Mary's and Tottenham Hotspur.
Wolves 'learn' asking price for Clarke-Salter
Meanwhile, according to Football Insider, Wolverhampton Wanderers have been told that they will need to come up with an offer of £10m to sign Jake Clarke-Salter from Queens Park Rangers.
Wolves are in the market for a replacement for Max Kilman, having allowed the centre-back to make the move to West Ham United during the current window.
Clarke-Salter has allegedly been identified as a leading target, and QPR have reportedly signalled that they would be open to a sale for around £10m.
Celtic, Crystal Palace, Burnley and Bournemouth have also all been credited with an interest in the 26-year-old, who started his professional career at Chelsea.
Clarke-Salter was an important player for QPR during the 2023-24 campaign, making 33 appearances in the Championship, scoring once and providing one assist in the process.
Rennes 'step up' pursuit of Kamara
Elsewhere, Rennes have allegedly offered a contract to Glen Kamara, as the French outfit step up their attempts to sign the Leeds United midfielder this summer.
The 28-year-old's future at Elland Road is currently the subject of much speculation despite the fact that he has another three years left to run on his contract.
Kamara arrived at Leeds from Rangers last August, and he featured on 42 occasions for the Whites during the 2023-24 campaign.
According to Football Insider, Rennes have offered the Finland midfielder a contract, but Leeds will not allow him to leave unless a 'significant' bid is submitted.
Kamara came through the youth system at Arsenal before moving to Dundee in 2017, and he then made the switch to Rangers in 2019, featuring on 193 occasions for the Scottish giants in all competitions.