David Moyes is a long-term admirer of the England star, who is expected to leave his boyhood club this summer.
The Hammers want at least two central midfield signings, with Argentine playmaker Manuel Lanzini also set to depart.
West Ham would ideally like a marquee addition to replace Rice, but filling the void is unlikely to be a one-man job, particularly given his outstanding fitness record. The 24-year-old has missed just one League game this season.
The Hammers boss values Ward-Prowse’s renowned set-piece delivery, having grown frustrated at his side’s lack of goals from such situations this season.
Ward-Prowse joined the Southampton academy at the age of eight and remaining in the Premier League will be essential to his hopes of making the England squad for next year’s European Championship.