The 28-year-old is expected to return to England's top flight following the Baggies' relegation, though Albion are understood to be unwilling to part with one of their star players on the cheap, given he is contracted to the club for another three years.
The Hammers are said to be among a number of clubs keeping an eye on any developments, with Crystal Palace, Bournemouth and Burnley all in the hunt to add Rodriguez to their respective forward lines this summer - as per Sky Sports.
The report claims West Brom will only consider selling should they receive offers of more than £20m, only a year after he joined the club in a £12m deal from Southampton.
Rodriguez was one of the Baggies standout performers in a relatively lacklustre season, scoring 11 goals in a campaign that saw Albion relegated back to the Championship following an eight-year stint in the top flight.
The Hammers interest in the 28-year-old comes as the club's new manager Manuel Pellegrini looks to restore an acceptable level of talent in the striking department, particularly with the futures of Javier Hernandez and injury prone Andy Carroll looking increasingly uncertain.
West Ham will be buoyed by Rodriguez's recent record with injuries as well as his goalscoring form, with the former Southampton ace having missed just one league fixture this season.
Rodriguez could be tempted to link up with his hometown club Burnley though, should Sean Dyche choose to make an approach ahead of the Clarets maiden Europa League campaign next season.