The person spoke to The Associated Press Thursday on condition of anonymity because the team and the quarterback have not publicly discussed the new offer. The person did not provide details about financial changes in the new offer, but added, ''as always, there is a lot to get through and no deal is imminent.''
The Saints have placed their one-year franchise tag on Brees, barring him from negotiating with other teams. Brees has said he wants to remain in New Orleans but has skipped voluntary practices and this week's minicamp while holding out for a long-term deal.
If a deal were to get done over the weekend, Brees would be able to attend the Saints' final set of voluntary offseason practices next week.
Brees was in New Orleans on Thursday, but not for contract negotiations.
He attended a City Council hearing to support a zoning ordinance required for a Jimmy Johns sandwich shop he will open not far from his New Orleans home in the city's historic Uptown district.
The council voted unanimously in favor of the variance, but Brees declined to be interviewed or make any comments about football as he was escorted to a waiting car outside City Hall.
Brees boarded a flight back to San Diego Thursday afternoon.
Brees, who joined the Saints as a free agent in 2006, has during the past six seasons completed more passes (2,488) for more yards (28,394) and more touchdown passes (201) than any other quarterback in the NFL. His 67.8 percent completion rate spanning the past six seasons also tops the league.
Arriving in New Orleans the season after Hurricane Katrina struck, Brees has become arguably the most popular sports figure in the city's history. In addition to leading the club to its lone Super Bowl title in the 2009 season, Brees has become a civic pillar of the community. He and wife Brittany have worked through their Brees Dream foundation to sponsor about $8 million in projects primarily aimed at improving the lives of children in the area through the refurbishing of schools or renovation of athletic fields and facilities. His foundation also has supported the arts, along with hospital facilities for cancer patients.
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.
He led the Saints to a 13-3 regular season record and second NFC South Division title in three seasons. New Orleans defeated Detroit in the first round of the playoffs before falling in the final seconds of their second-round game at San Francisco.
This offseason marks the first time Brees, 33, has had the opportunity to negotiate a major, long-term deal with the leverage provided by a string of injury-free seasons highlighted by a slew of individual and team statistical records in addition to a championship.
He commanded only second-round money when he was drafted by San Diego, and the Chargers placed the one-year franchise tag on him when that deal ran out in 2005. The right-handed Brees then had an injury the following season that required major surgery to his throwing shoulder, and which left him with few major offers during the 2006 offseason.
He wound up signing a six-year deal with New Orleans worth about $60 million, which left him playing for well below what the top quarterbacks in the league earned during the past few seasons.
Brees is now expecting to be paid in the range of the league's best quarterbacks, and Peyton Manning, who is three years older than Brees and sat out last season while recovering from neck surgery, recently signed a contract with Denver paying him $19.2 million a year.
Brees has until July 16 to agree to a new long-term deal. Otherwise he will have to either play under the franchise tag that would pay him a little more than $16 million for one season, or sit out.
Getting Brees back with the team would help restore a sense of normalcy after an offseason plagued by the NFL's bounty scandal, which resulted in the season-long suspensions of head coach Sean Payton and linebacker Jonathan Vilma, among other sanctions.
So far, Chase Daniel has worked as the No. 1 quarterback during six voluntary offseason practices and this week's three-day minicamp that ended Thursday. The Saints also gave veteran Luke McCown a tryout during minicamp and decided on Thursday to sign him as well, meaning he'll likely be back for next week's offseason practices and possibly training camp.
Saints players and coaches have said it is odd not having Brees around, but added that they are confident his absence won't unduly hurt the team as long as he's back by training camp.
Assistant head coach Joe Vitt, who has been tapped to handle head coaching duties this season, except while serving his own six-game suspension resulting from the bounty probe, said he has been talking with Brees regularly and asking the quarterback for input on training camp, which opens in late July.
''Nobody is going to come in better shape than Drew,'' Vitt said earlier this week. ''Drew is working extremely hard right now. He's mastered the system. (Brees' contract) will get done with both parties getting together. ... They are both working hard at it.''
Notes: The Saints also signed running back Joe Banyard after the completion of minicamp. Like McCown, Banyard was with the Saints on a three-day tryout this week. ... Starting RT Zach Strief wound sat out a fourth-straight practice on Thursday after rolling his right ankle in the opening practice of minicamp. He said the injury is minor, that he rested as a precaution, and that he expects to practice next week. ... RB Chris Ivory sat out a second straight day with what Vitt has said is a minor hamstring pull. ... The Saints announced that they will hold training camp, as usual, at their suburban New Orleans headquarters and that it will open on July 27.