The former fifth-round draft pick from Penn State already is drawing plenty of interest on the free-agent market, according to his agent, Chris Cabot.
Cabot said Monday that he has had discussions with the Packers and seven other NFL teams about Quarless since the free-agent negotiating period opened on Saturday.
Until 4 p.m. ET on Tuesday, only the Packers can sign Quarless. If a deal can't be reached by then, Quarless would be free to sign with another team.
The 6-foot-4, 252-pound Quarless is coming off a season in which he set career highs in receptions (32), yards (312) and touchdowns (two). He became the Packers' starting tight end after Jermichael Finley's season-ending neck injury on Oct. 20.
In perhaps the best display of what he could potentially become, Quarless had back-to-back games with six catches for 66 yards and a touchdown against the Atlanta Falcons in Week 14 and Dallas Cowboys in Week 15.
Quarless missed the entire 2012 season because of a knee injury he sustained late in 2011, and the Packers believe he still has room for growth.
"I think he's got more in the tank, and we talked about that throughout the season," Packers tight ends coach Jerry Fontenot said shortly after the season. "I think that he's made strides in being a better receiver and understanding where he needs to be on the field at any given point. As with anything, you always work on consistency."
Cabot did not name any of the seven other teams who have inquired about his client, but among those believed to be interested in Quarless are the New York Giants, where former Packers assistant coach Ben McAdoo is the new offensive coordinator. McAdoo was Quarless’ position coach in 2010 and 2011 before moving over to quarterbacks coach.