Barnes went under the knife on Wednesday after rupturing an Achilles in the Rams' pre-season friendly against Stoke at the weekend.
The 20-year-old, who has one-year remaining on his contract at Pride Park, spent almost 10 months on the sidelines with a serious knee injury last year.
Clough said: "He will certainly be out for a few months, but it depends on the severity of the rupture.
"We will know how bad it is after the operation but we'll just try and get everything sorted when he comes back at around Christmas time or hopefully somewhere round there.
"It's a blow for him because he will have been out with injuries for the best part of two years almost."