Browns general manager John Dorsey said on Thursday he has been trying to contact Bryant, but that the former Dallas Cowboys wide receiver was not returning his calls.
Bryant took to Twitter in response and said he will be there sometime next week, perhaps Thursday.
"For real ... I wouldn't mind playing for the (Browns) ... I just want to be right first," Bryant wrote in a tweet.
Two hours later Bryant tweeted again: "Starting my visits next week ... I'm coming to the Land to see your Mr Dorsey."
Earlier, Bryant tweeted: "I'm going to play ... just whenever I get ready."
NFL Network Insider Ian Rapoport reported on Friday: "He's been waiting for the right fit, so this visit could be as important as whatever contract offer may come. They have to mesh well."
The Browns are looking for help at wide receiver while Josh Gordon remains away from camp because of health issues. Gordon told head coach Hue Jackson on HBO's Hard Knocks this week that he would return "soon."
However, Dorsey said on Thursday that there is "no timetable" for Gordon's return.
Cleveland also traded wide receiver Corey Coleman to the Buffalo Bills hours before learning that fourth-round draft pick Antonio Callaway was cited for marijuana possession.
Callaway claims the marijuana discovered in his car during a traffic stop was not his, but is waiting to if the NFL takes an action. The rookie caught a 54-yard touchdown pass in Thursday's preseason opener.