After over a decade at Manchester United, the 29-year-old midfielder will be leaving Old Trafford when his contract expires on June 30.
Lingard enjoyed a successful short-term loan with the Hammers during the second half of the 2020-21 campaign, scoring nine goals and providing five assists in 16 Premier League appearances.
According to Sky Sports News, West Ham have made an offer to Lingard's representatives over recruiting the player on a free transfer this summer.
Newcastle United have also been linked with Lingard in the past but the Hammers are believed to be the frontrunners for the midfielder's signature.
West Ham have already announced the signing of centre-back Nayef Aguerd from Rennes for around £30m, while goalkeeper Alphonse Areola is believed to be close to finalising a permanent transfer to the London Stadium after spending last season on loan from Paris Saint-Germain.
David Moyes is keen to bolster his squad ahead of the 2022-23 campaign, with Chelsea striker Armando Broja and Southampton captain James Ward-Prowse also understood to be on his list of targets.