Sam Allardyce – whose Hammers job in in jeopardy amid fan unrest – is said to be plotting for next season already and has made securing a forward to support Andy Carroll a priority.
For much of the season the London club were without a top-quality goal-getter as the burly Geordie missed lengthy periods through injury and suspension.
Carlton Cole was re-signed after being released, while Marco Borriello arrived on loan, but neither were able to provide the sort of cover needed.
Now Allardyce is looking at Matri, who has spent the letter half of the campaign on loan at Fiorentina.
The 29-year-old is believed to be surplus to requirements at Milan, and will be available for the right price – believed to be a minimal sum – this off-season.
Matri possesses the experience ‘Big Sam’ craves, having represented Italy at senior level while claiming two Serie A titles at Juventus.
The silky striker has spent his entire career in his homeland, but West Ham are hopeful that they can lure him to the Premier League despite interest from other clubs.