The 24-year-old was recently reported to be a target for new West Brom manager Alan Irvine, but it now appears he has been lured to Upton Park.
Ipswich are believed to have already turned down a bid from relegated Fulham for the full-back, who has made over 130 appearances for Town after joining from Tranmere in 2011.
Hammers boss Sam Allardyce is understood to have had a bid for Norwich defender Martin Olsson rejected, forcing him to turn his attention to the highly-rated Cresswell.
Cresswell signed a new contract with the Championship club last December, but Ipswich boss Mick McCarthy admitted on Monday they would consider offers if the price was right.
He said: "There is interest, I can't deny that, as he is a good player, but if there are any bids, I won't be telling you.
"He was getting the plaudits last season for being the best left-back in the Championship and at some stage, if someone comes in and the deal is right, then it becomes an irresistible force.
"He is under contract, but that becomes a negative as the player sees there has been a bid for him that everyone else seems to think is fair.
"He may feel he's done his time and want to move to better himself. If that's the case, it's hard to stop.
"If he does go I will shake his hand, thank him for his efforts and crack on and bring someone else in, that's how it works. It would have to be the right bid though."