The Rams boss said the offer, believed to be in the region of £1million, was "swiftly rejected" as it fell short of their valuation of the player.
The FA Cup holders could return with an improved bid but Clough insists he is under no pressure to sell the 25-year-old right-back, who has also been linked with Cardiff and West Ham.
Brayford has a year remaining on his existing contract at Pride Park and the club have held initial talks about a new deal.
"We had an offer that was swiftly rejected," said Clough. "They weren't particularly close. If it was structured in a different way, they may not have been too far away but the way it was structured wasn't right.
"Our chief executive deals with that side of things and he was in agreement straight away that we should turn it down. If an offer came in that did meet our valuation, it would then be up to John.
"But there's no indication from the player that he wants to leave - quite the opposite. And we're not under any pressure to accept a bid.
"If he doesn't fancy it, we will sit down and talk about a new deal any time he wants. We've had initial talks with him but we said, 'let's just see what happens over the next few weeks'.
"There was previously some interest from the league above as well and, at 25, he's earned the right to see what's out there - but we hope he'll remain with us.
"What we can't afford to do with players is let them go for nothing in 12 months' time. He's got 12 months to go, he's been brilliant for us and he's entitled to go and have a look around and see what's on offer."
- Sports & Recreation