The Hammers are currently looking for a manager to succeed the departing Sam Allardyce.
Sevilla coach Unai Emery is the Premier League outfit’s top target, but it seems that they may struggle to convince him to move to the Boleyn Ground this summer.
According to The Telegraph, the former Valencia coach has been approached by Italian club Napoli, with Sevilla also keen for him to stay.
It has been reported by the British newspaper that Monk is on the radar of West Ham, despite the fact that he is already at Premier League rivals Swansea.
“Swansea City manager Garry Monk is on the West Ham list and is yet to sign a new contract at the Liberty Stadium,” states the report.
This is not the first time that Monk has been linked with the managerial job at West Ham. During the 2014-15 season, the 36-year-old former defender had to publicly distance himself from the Hammers.
"West Ham have a fantastic manager in place," BBC Sport quoted Monk as saying in April. "For them to be looking elsewhere - if they are - I think they've already got the manager there who's had a fantastic season.
"I look at that type of speculation as a compliment to the way myself and the team have worked this season, showing we're progressing and going in the right direction."
Monk has done a truly remarkable job as manager of Swansea, but it is unlikely that West Ham will be able to prise him away from the Liberty Stadium this summer.
The former Southampton and Barnsley defender seems settled at Swansea, and there does not appear to be a reason for him to leave and take charge of a new team right now.