Bryant has yet to sign with a team after getting released by the Dallas Cowboys in April.
This week, NFL Network's Michael Silver reported the Browns were still "very interested" in signing Bryant.
Bryant is coming off three down seasons after registering at least 1,200 yards and 12 touchdowns in three straight campaigns from 2012-2014.
The three-time Pro Bowler reeled in 69 receptions for 838 yards and six touchdowns last season. He hasn't reached the 1,000-yard mark since 2014.
Cleveland's top two options at wide receiver currently are trade acquisition Jarvis Landry and Josh Gordon, who rejoined the team last week following a leave of absence. Rookie fourth-round pick Antonio Callaway and third-year man Rashard Higgins are also in line for expanded roles.
If the Browns sign Bryant, they could play him and Gordon on the outside, with Landry settling into his familiar slot position.