However, they are hesitating over making a concrete bid because of concerns over the state of his right knee.
The World Cup winner missed part of last season and was far from his best as a result, and it was also the root of multiple injuries.
Crystal Palace also want the 27-year-old, but are stalling over making a move because of the same reason.
Hammers are looking to beef up their backline ahead of the new season, and with Pablo Zabaleta entering his last year at the club, Sidibe will represent a great replacement and addition capable of providing cover and competition for Ryan Fredericks.
Monaco want around £14 million for the France international, and West Ham are open to meeting with their demands as they know the player has the talents and is now entering his peak.
However, his injury woes understandably remain a cause of concern, and it will be interesting to see what their final decision is in the coming 10 days.
Manager Manuel Pellegrini is looking to lead the London Stadium outfit to Europe at the end of the new season, and has to get a squad capable of finishing as best of the rest.
A player like Sidibe brings quality and experience to the table, but given their experience with Andy Carroll and seeing how his former teammate Benjamin Mendy has been unable to play for Manchester City because of a similar injury, taking a gamble and going ahead to snap him up isn’t very enticing.