The report states that the Ivory Coast international is set to sign for the Men in Red if he can agree terms with the MSL.
Chicago Fire have offered the experienced striker a €2.4 million per season deal. Drogba has also received offers from Qatar.
Drogba spent two seasons at Galatasaray and scored 20 goals in 53 games before joining Chelsea in the summer.
The Ivory Coast international won the Süper Lig, Turkish Cup and reached the quarter-final of the Champions League while at the Lions.
Drogba provided two assists and seven goal in 40 appearances for Chelsea last season.
The former Galatasaray striker is currently a free agent after his contract with Chelsea expired in the summer. Drogba has won the Premier League on four occasions.