According to a Sun on Sunday source, the Liverpool-born 28-year-old is keen on a return to his hometown – although he has previously admitted to being a boyhood fan of the Reds.
Allardyce bought Cresswell to West Ham from Ipswich in 2014 for a transfer fee believed to be £4.28m signing an initial five-year contract. After his debut season, Cresswell signed a new five-year contract and again in 2016 Cresswell signed a one-year contract extension to keep him at the Hammers until 2021.
Cresswell has made 150 appearances as a Hammer in all competitive games since joining.