The Conference champions have apparently snapped up the former England under-21 international but the transfer is still subject to personal terms.
The Bantams could be in line for a windfall as part of the deal if Chester are promoted again next season.
The club say they would also be entitled to a cut from any subsequent transfer.
According to reports earlier this week, Chester City signed the player on a free transfer.
The Bantams quickly rejected those claims, insisting they were only prepared to negotiate a fee for Branch.
Meanwhile, two other players have sealed moves away from Valley Parade. Midfielder Robert Wolleaston is making the switch to Third Division Oxford United with New Zealand goalkeeper Mark Paston heading for Walsall.