Honda has been linked with the Hammers for the past few weeks, although Milan have shown no apparent desire to sell him in the upcoming transfer window.
However, the Japanese international - a holder of 73 caps - has found himself fall out of favour for the Rossoneri, with Sinisa Mihajlovic relegating him to a substitutes role.
He has not started a league game since the middle of September, and has totalled just 17 minutes of action in the last six games.
Due to his changing situation, Honda has asked to leave Milan in January, but CEO Adriano Galliani has expressed no intentions to sell him whilst Milan suffer something of an injury crisis.
Milan do want to cash in on Honda soon, so as to ensure they do not lose him for free when his contract expires in the summer of 2017, but Galliani would prefer a sale next summer.
However, if the Hammers put in a bid which is acceptable for Milan, Galliani could change his view on the sale of Honda in January.