The former Chelsea ace comes off contract in June and won't be offered new terms at season's end.
Essien has been linked with lucrative offers from the Middle East and MLS since the turn of the year.
However, Sportmediaset says West Brom have made a firm approach to Essien's management about a return to England this month.
The Rossoneri aren't expected to stand in his way should he accept the Baggies' offer.
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