Byrd is seeking a deal worth at least $9 million per season, according to ESPN Miami Dolphins reporter James Walker. However, some early signings Tuesday may not work in Byrd's favor.
The Cleveland Plain Dealer reported that former Browns safety T.J. Ward signed a deal worth $5.5 million per season with the Denver Broncos. Along with Byrd, Ward was considered among the top safeties on the free-agent market.
The Saints had $2.1 million in cap space Monday but could restructure contracts in order to afford signing Byrd.
Meanwhile, Bills general manager Doug Whaley said Tuesday that the "lines of communication" remain open between the Bills and Byrd, a three-time Pro Bowler.