A number of injuries throughout the last campaign saw Antonio fall out of favour under David Moyes, and many expect that the recent appointment Manuel Pellegrini won't help the Englishman's chances of breaking back into the West Ham first team.
The club are understood to be open to offers for the 28-year-old this summer, as they look to rebuild their squad ahead of the new season.
The Guardian has reported that Crystal Palace will look to test the strength of West Ham's resolve this summer, with the Irons set to be busy elsewhere in trying to retain the services of Marko Arnautović - amid interest from Manchester United and Everton.
Crystal Palace are themselves braced for a flurry of transfer interest heading in the direction of Wilfried Zaha. The likes of Arsenal, Tottenham and Liverpool have all been previously linked with a move for the Ivory Coast international.
West Ham will be eager to recoup most of the £8.5m they paid for Antonio back in 2015 ahead of their squad overhaul this summer.
The Hammers are understood to be keen on resigning João Mário from Inter on another loan deal this summer, while the club are stalling on a £17m deal that would see Barcelona defender Marlon Santos move to east London.