The Mirror says Redknapp is keen to bring in the Boro skipper in order to strengthen their midfield.
It is believed Boro would be interested in selling their star, should the right offer arrive.
Boro boss Gareth Southgate has England Under-21 kid Adam Johnson as cover, should Downing leave the club in January.
Liverpool and Aston Villa have also been linked with Downing in recent months.
