With the Dons languishing in mid-table and Bamford the second top-scorer in the division, the Blues may feel the youngster's development would better suit a stronger league.
The Derby Telegraph reports that Dons boss Karl Robinson is resigned to losing the striker in the near future, while quoting Rams' boss McClaren on the possibility of signing Bamford: 'We are having a little look around.
'What we have done is we have gone for younger players, the likes of Andre Wisdom and Michael Keane, and even Simon Dawkins, who I keep thinking is only 18 or 19, and he is 26. So, we have gone for younger players from the top teams and we want that quality.
'We are looking for youngsters again to add to the squad and improve the quality we have got.'
Bamford started out at Derby's Midland rivals Nottingham Forest, before joining Chelsea for £1.5 million in 2012.
Derby are sitting in second place in the Championship, and may face competition from others in the division if Bamford does indeed move up a division.