The 27-year-old - who worked under Black Cats boss Sam Allardyce at Upton Park - has made close to 250 appearances for the Hammers.
But he could find himself surplus to requirements as boss Slaven Bilic looks to resurrect his squad, after a seventh-place Premier League finish.
Crystal Palace are also reportedly interested in Basildon-born Tomkins.
Allardyce is looking for defensive recruits as part of his summer rebuilding plans.
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Meanwhile, Allardyce has opened discussions with chairman Ellis Short over Sunderland’s summer transfer kitty.
The pair are understood to have met yesterday and will continue their talks today after Sunderland’s summer planning has largely been on hold while there has been such uncertainty over their top flight status.
The transfer kitty is not the only issue on the agenda, with Sunderland’s wage bill around the Â£70million mark and still bloated by several players who are due to return to the Stadium of Light after completing their loan moves.
Once Allardyce and Short have finalised the details of Sunderland’s budget, then the process can begin of more detailed talks over permanent moves for the club’s loanees or new contracts for those already on the books.
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And finally, one winger who is definitely part of Allardyce's pre-season plans is youngster Rees Greenwood.
Greenwood has been the stand-out figure for Sunderland Under-21s this season and had trained regularly with the first-team squad for several months.
The England Under-20s winger was given his Premier League bow in the final day draw at Watford.
Greenwood said: "I would like to think I can stay in and around the first team picture next season, but that’s not up to me.