Weeden's deal is worth $1.23 million over two years and does not include bonus money, according to the source.
A first-round pick of the Browns in 2012, Weeden posted a 5-15 record before his release last week. He completed 438 of 784 passes for 5,116 yards with 23 touchdowns and 26 interceptions in two seasons. He nearly beat the Cowboys in 2012, throwing a go-ahead touchdown pass late in the fourth quarter only to see Romo direct a game-tying drive at the end of regulation and the winning drive in overtime.
Since coach Jason Garrett arrived in 2007, the Cowboys have carried veteran backup quarterbacks in Brad Johnson, Kitna and Orton. Weeden, 30, does not have the same experience, but joining the Cowboys would give him a chance to rebuild a career that took a big hit with the Browns.
Orton is under contract through 2014 as part of a deal he signed in 2012.
The Cowboys believe Orton, 31, will continue to play in part because of a $3.25 million base salary and the fact he would have to repay the team $3 million of a $5 million signing bonus should he decide to retire.