Hammers manager David Moyes admitted last week that there is a "good chance" Rice will leave the London Stadium this summer amid interest from a host of top Premier League clubs chasing Champions League football.
West Ham value Rice in the region of £100m and they are under pressure to sell the 24-year-old this summer to avoid losing their star man for free when his contract expires in June 2024.
Arsenal are believed to be leading the race for Rice's signature at present, while Manchester United have also been credited with an interest, although Manchester City allegedly have 'concerns' over a potential swoop for the England international.
Should West Ham agree to cash in on Rice this summer, the Evening Standard reports that the Hammers are ready to take advantage of Southampton's relegation from the Premier League and table a bid for Ward-Prowse.
The report adds that Moyes is a long-term admirer of the 28-year-old and he considered a move for the Saints skipper last summer before splashing the cash on Lucas Paqueta and Flynn Downes, recruiting the former for a club-record fee.
West Ham are said to be targeting a marquee addition to replace Rice, with Ward-Prowse one name on their radar, and they will also target another midfielder with Manuel Lanzini expected to seek pastures new this summer.
Ward-Prowse has spent his entire senior career at Southampton and is regarded as a club legend on the South Coast, contributing with 54 goals and 52 assists in 407 games across all competitions since his debut back in 2011.
However, speculation over his future at St Mary's is expected to intensify as a result of Southampton's relegation from the Premier League, with West Ham set to face stern competition for the midfielder's signature.
Man United are understood to have added Ward-Prowse to their summer shortlist, while the likes of Tottenham Hotspur, Liverpool, Newcastle United and Aston Villa have also been credited with an interest.
Following confirmation of Southampton's relegation after last weekend's 2-0 defeat to Fulham, Ward-Prowse acknowledged that he is "not thinking too far ahead" about his long-term future at St Mary's.
Ward-Prowse, who has built a reputation as a set-piece specialist during his career, is Southampton's top scorer this season with eight Premier League goals in 36 appearances, and he has also chipped in with three assists from midfield.
The 11-cap England international signed a new contract at St Mary's last summer until June 2026, and The Mail reports that it does not include a relegation release clause.