Gomez spent last season on loan at Turkish side Besiktas netting 26 times in 33 appearances over the course of the campaign.
Following such impressive displays, reports have suggested that Hammers boss Bilic has been keeping a close eye on the performances of Gomez at Euro 2016. The Croatian has been in France for the tournament, working as a pundit for ITV's coverage of the tournament.
Thanks to Slaven Bilic for his part in @ITV#EURO2016 coverage so far. As agreed pre-tournament, Slav is now returning to #WHUFC Hvala!
Gomez has enjoyed somewhat of a renaissance in recent years returning to the national team in 2015 after a 436 day absence.
The big striker will have impressed the watching Bilic, continuing his sparkling domestic form in France at Euro 2016.
After Germany lacked a real goal threat in their two opening games against Ukraine and Poland, with Mario Gotze deployed in a false-9, Gomez returned to the starting eleven against Northern Ireland.
Gomez's goal proved the difference in Die Mannschaft's final group game as Germany narrowly defeated plucky Northern Ireland by a goal to nil. The 30-year-old went on to score in the German's rout over Slovakia in the last-16 tie on Sunday.
If Gomez is to move to East London, it is reported by CNN Turk that Besiktas will benefit from 40% of Gomez's transfer fee. Although manager Bilic remains confident having spent two-years at Besiktas before taking charge at West Ham.
The German is not the only front man on Bilic's wish list though, with reports linking the move for Lyon's Alexandre Lacazette and Marseille's Michy Batshuayi.