The 25-year-old has entered the final year of his contract at Upton Park and is yet to sign an extension.
The Hammers will be keen to put fresh terms to a product of their famed academy system, but as yet no agreement has been put in place.
Fulham have decided to test the water and have made contact with their Premier League rivals regarding a possible deal.
Noble is a proven top-flight performer and was instrumental in guiding West Ham to promotion last season.
As an East End native, he has strong ties to the Hammers and has made over 200 senior appearances for the club.
He has started both of West Ham's Premier League fixtures so far this season, but it remains to be seen how hard Fulham push to bring in a man who would help to fill the void created by Moussa Dembele's proposed switch to Tottenham.