Winston is expected to become an unrestricted free agent after five years with the club. The team can sign it at any time but has not done so.
In addition to Barrett, Jason Pierre-Paul, Carl Nassib, Ndamukong Suh, Beau Allen and Rakeem Nunez-Roches are all ready to become free agents.
Barrett is optimistic about his stay in Tampa.
"I don't know if it's going to happen before [March] but I like this confidence [coach Bruce Arians] gave me. I'm just ready to do it and ready to start building my life in Tampa and be there for the long haul, "he said last month in Orlando, Florida, at the Pro Bowl.
The Arians declined to comment on Winston's decision to have LASIK eye surgery recently, despite discussions about Winston's myopia at the end of the season.
"I'm not talking about guys' personal affairs," said Arians.
A source close to the situation told ESPN that the choice was Winston's and was not made at the insistence of the team.
When asked if he felt closer to having more clarity about his quarterback situation, the Arians replied, "No, because you don't know who is available. Are you just sitting and waiting to see that there is someone available? And is there a better option?
" This is the problem. You have about three days to decide - legal forgery [period]. Is someone else going to let go? What are the commercial values? It's really dull right now because you don't really have the answers. "