"I'm an eternal optimist, so I'm still very confident a deal will get done by Monday," Brees said at the ESPY Awards on Wednesday night, per the Times-Picayune.
"I think I've been around this league long enough to understand how this works at times. In a lot of cases, it does go down to the wire, and you need a stopping point in order to bring everyone together and make sure it can get done. That stopping point is Monday, so let's hope we get something done by then."
Brees plans to sit out training camp if a multi-year deal isn't reached, and the sides are believed to be about $10 million apart in guaranteed money, according to the paper.
Brees was given the Saints' exclusive rights free agent tender in March, kicking off more than three months of sometimes tense talks.
Brees faces the July 16 deadline -- the date for all franchise players to sign extensions or play the 2012 season under the franchise marker -- with the alternative being a one-year salary of $16.371 million, the franchise tender for his position.
Brees, as argued by the NFLPA in June, won a grievance the determined he will be owed a 44 percent raise up to $23.785 million if he's again named the Saints' franchise player in 2013.
While Brees is a three-time franchise player, only two of the instances came with the Saints. The other, in 2005, was his final season with the Chargers. Brees would have been owed a 20 -percent raise up to about $19.595 million had he lost the grievance.
Brees is coming off of a record-setting season in which he broke Dan Marino's record for passing yards with 5,476 and completed 71.2 percent of his passes, breaking his own record (70.6 in 2009). Brees and the Saints set a record for total offense, gaining 7,474 to beat the 2000 St. Louis Rams' mark of 7.075.
The Saints are thin at the quarterback position. Backup Chase Daniel (hairline fracture in his right thumb) was injured in practice on June 13 and might not be full strength at the start of training camp.
If Brees doesn't report, that could leave the Saints' first-team offense under the direction of Sean Canfield or veteran backup Luke McCown, who signed in June.