However, the deal went cold after his fee increased, and City instead went for Jamie Forrester, Ben Burgess and Danny Allsopp.
When Branch was released at the end of his contract by Wolves the rumours started up again, but Peter Taylor decided not to add him to his ranks.
The 25-year-old former Everton player has now signed a two-year deal with Bradford City. Manager Nicky Law also recently secured the services of fellow ex-Tiger Dean Windass.
Meanwhile, Taylor is set to unveil Jason Price as a Hull City player after the winger left Tranmere. Price joins former Rovers team-mate Richard Hinds, as well as fellow new signings Alton Thelwell, Andy Dawson and Danny Allsopp in pre-season training with the City squad.