Byrd's current deal runs out during the off-season and attempts to agree an extension were unsuccessful last season, with the 27-year-old eventually playing under the franchise tag for the 2013 campaign.
However, Brandon is keen to start afresh in new talks over a contract for the three-time Pro Bowler and is hopeful that this time they can be more successful.
"It's a process and there's no carryover. You start over again and you try to get something done. We just weren't able to come up with a solution that worked for both parties, so you reset it and start again," Brandon told The AP.
"We all are hoping to get a deal done and we're going to do the exact same thing this time, work our tails off to try to get another one done."
Byrd recorded a joint team-high four interceptions as well as a sack and a forced fumble last season, despite missing the first five games of the campaign.