The Boro boss is a huge fan of the versatile midfielder, having worked with Hargreaves while with the England national team.
Hargreaves`s current deal is set to expire at the end of the season, and he has so far failed to consider signing an extension.
The youngster has made no secret of his desire to play in England, and Bayern boss Ottmar Hitzfeld said on Thursday that he may not be able to keep Hargreaves in Germany.
McClaren has been alerted to the situation, and confirmed he will monitor the developments closely.
He said: "He is a player who I admire a great deal, especially after working with him with England.
"If players of his quality become available, we would be mad not to be interested in him."
