Madrid are ready to let Drenthe, 21, who can play at left-back or left midfield, go after sealing a £20million deal for Pompey ace Lassana Diarra.
The Dutch Under-21 international has been recommended to Adams by Portsmouth's first-team coach Johnny Metgod, who worked with Drenthe at Feyenoord.
Drenthe joined Madrid in June 2007 after being named player of the tournament in the Under-21 European Championships but has struggled to cement a regular first-team place and is not in the immediate plans of new coach Juande Ramos.
Portsmouth are also on the verge of re-signing midfielder Gary O'Neil from Middlesbrough for £4m.
O'Neil made 174 league appearances for Pompey before being sold to Boro in the summer of 2007 but is unsettled in the North East.
Adams, who has won only three of his 14 games in charge at Fratton Park, said: "January is always a big month."
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