The FA Cup winners, relegated from the Premier League last season, have tabled an offer in the region of £1million for the 25-year-old.
The approach has been rejected as it does not meet Derby's valuation of the player, who has a year remaining on his contract at Pride Park.
The Rams are keen to extend Brayford's deal beyond next summer, but manager Nigel Clough does not want to risk losing him on a free transfer and the club may consider an improved offer.
Brayford joined Derby from Crewe in May 2010. He previously played under Clough at Burton and and has been strongly linked with a move to Premier League newcomers Cardiff.
- Sports & Recreation
- Premier League
- Nigel Clough