Gimenez, 24, was prolific in the Eredivisie with Feyenoord but has been an undeniable flop since joining AC Milan in January.
The Mexican forward has only managed to score seven goals in his 30 games across all competitions and has fallen out of favour under Max Allegri.
Now, according to Sportal.It, the Serie A giants have given up hope on Gimenez and are looking to secure a swap deal with West Ham for Fullkrug.
Fullkrug has hardly been a hit in East London with three goals in 27 games since joining last summer and is currently sidelines yet again with a muscle injury.