The Hammers are looking to bolster their side as they look to challenge for a place in the top half of the standings next season and they are looking to the relegated sides for possible new recruits.
Shawcross is not the only Stoke star whom West Ham have been linked with.
Others include Xherdan Shaqiri, Jack Butland and Joe Allen.
The 30-year-old Shawcross has been with the Potters for over a decade and is very experienced at Premier League level.
He has over 400 appearances to his name and Stoke will not be willing to let him go easily.
He could provide valuable leadership to the Potters next season as they look to return to the Premier League at their first attempt back in the Championship.
It remains to be seen whether he will opt to extend his stay in the Premier League or remain with the Potters and try to establish himself as a legend with the club.
Manuel Pellegrini’s side will need to find a replacement for James Collins who has been released.
They may also need to find a replacement for Winston Reid as the New Zealand international has been linked with a possible move away from the London Stadium this summer.