The Hammers need to sign a back-up goalkeeper this summer because Adrian is leaving the London Stadium at the end of his contract.
Caballero is out of contract at Chelsea next month and the 37-year-old Argentinian will be available on a free if he is not given a new deal.
West Ham manager Manuel Pellegrini worked with Caballero at Manchester City and the pair could be reunited.
Caballero made nine appearances for Chelsea this season, and kept a clean sheet in their 2-0 win against Tottenham in February after Kepa Arrizabalaga was dropped for refusing to be substituted in the Carabao Cup final penalty shootout defeat to Manchester City.
Chelsea signed Caballero on a free from Manchester City in 2017 and at the end of last season they took up the option of a further 12 months on his contract.
West Ham are exploring their goalkeeper options, with Adrian departing after six years at the club and more than 100 appearances.
Pellegrini wants an experienced goalkeeper to provide competition and back-up for Fabianski, who was named Hammer of the Year after a brilliant first season at West Ham following his £7million move from Swansea last summer.
West Ham’s 20-year-old academy goalkeeper Nathan Trott could compete for the No1 spot in years to come but he is likely to be loaned out next season to gain experience.