Bryson and Brayford both have another year to run on their current deals.
Bryson has been a key figure for the past two seasons. The 26-year-old Scot has made 85 appearances since arriving from Kilmarnock in 2011.
Clough told the Derby Telegraph: "It's a priority for us to get him tied down on a new deal.
"There are not too many Craig Brysons out there. We've been looking, but we've only found one in the last couple of seasons.
"With his attributes and qualities, we see him being integral to our plans for the future."
Speculation surrounded Brayford last summer and again during the January transfer window.
The 25-year-old was strongly linked with Cardiff, who have won promotion to the Barclays Premier League, while it has been reported that West Ham are interested in the right-back.
Clough added: "There was some interest in him in the last two windows, so I think he will probably want to have a little look and see what is out there."