The two sides, both of which believe they are making a fair offer, are facing a gap of $5 million per season for the first three seasons of the proposal, sources told Schefter.
Brees, who led the Saints to their only Super Bowl championship and last season broke Dan Marino's single-season passing yards record, is seeking a contract that averages $23 million per season for the first three years of his deal, sources told Schefter.
That's the same amount Peyton Manning's Colts contract averages the first three seasons.
But the Saints are offering Brees a five-year contract that averages $18 million a year, sources told Schefter, which is the average that Manning and New England's Tom Brady are due to earn over the life of their deals.
Brady's contract averages $18 million in the first three years, like the Saints' current offer to Brees.
If the two sides cannot make considerable progress from where they are now, the Saints will be forced to use their franchise tag on Brees by the Monday deadline.
Brees has said he is confident a deal will get done by the 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.
The 2011 offensive player of the year arrived in New Orleans in 2006, less than a year after Hurricane Katrina struck. Since then, Brees 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 divisional-round game at San Francisco.
Information from The Associated Press was used in this report.