If the two sides cannot make considerable progress from where they are today, the Saints will be forced to use their franchise tag on Brees by the Monday deadline.
If the Saints can get Brees signed to a long-term contract extension, they would have the flexibility to use their tag on another player. What makes this important is that the Saints have two other marquee free agents, wide receiver Marques Colston and Pro Bowl guard Carl Nicks.
Nicks took to Twitter on Monday to express his anxiety over possibly being tagged: "Man I don't know what to do? My anxiety is on a hundred thousand trillion!...march 5 is a big day!!??"
But as it stands right now, the team and Brees are struggling to strike the deal that many around the league expected to be done by now.
The 2011 offensive player of the year arrived in New Orleans in 2006, less than a year after Hurricane Katrina struck. Since then, he has passed for more yards (28,394) than any other quarterback in the NFL while lifting the Saints to new heights and simultaneously helping the region heal from Katrina's devastation.
In 2011, Brees set NFL single-season records with 468 completions, 5,476 yards passing and a completion percentage of 71.2. His prolific passing numbers helped the Saints set a new NFL high for total offensive yards in a season with 7,474.
The 33-year-old Brees led the Saints to a 13-3 regular-season record and second NFC South division title. New Orleans defeated Detroit in the first round of the playoffs before falling in the final seconds of its second-round game at San Francisco.
Last week, Brees told the New Orleans Times-Picayune he was confident he would reach a deal with the Saints. "There's no doubt in my mind we'll get a deal done," he said.
"I think there's been progress made," Brees said, when asked about reports that he and the Saints were far apart. "But just like everything it's a process. It takes time. It's not something that happens overnight."
Colston has had more than 1,000 yards receiving in five of his six seasons, coming up short of the mark only when he missed five games with a hand injury in 2008.
Last season, the 28-year-old Colston missed two games with a broken collarbone and finished third on the club in catches with 80, second in yards with 1,143 and tied for second in touchdown catches with eight.
Nicks, a two-time Pro Bowler at guard, was a member of The Associated Press' first-team All-Pro team in 2011, along with the Saints' other starting guard, Jahri Evans. That hasn't happened since 1953, with Detroit's Lou Creekmur and Dick Stanfel.
Adam Schefter is ESPN's NFL Insider. Information from The Associated Press was used in this report.