Quinn was flying to the Browns' headquarters Tuesday to sign the deal, a person within the league told The Associated Press on condition of anonymity because the contract's language was still being finalized.
The person said the deal, worth $20.2 million, with $7.75 million guaranteed, could reach $30 million over five years with incentives.
The deal was first reported by Foxsports.com.
Coach Romeo Crennel has coldly referred to Quinn as "the quarterback" and not by name during the holdout.
"He's pretty far behind," Crennel said last week. "We have a lot of offense, and we're putting it in every day. It takes a while to get this down and get caught up on it."
Quinn also missed the team's four-day rookie orientation before camp.
Condon proposed to allow Quinn to get a $5 million increase in the final two years of a potential five-year deal if he takes 55 percent of the snaps in any two of the first three years or 70 percent in any one of the first three. The Browns wanted to make the triggers tougher to reach.