Citing two league sources, the Times-Picayune of New Orleans reported Monday that a deal is near for Peterson, who visited the Saints two weeks ago.
The NFL Network reported that the sides are close to an agreement and that Peterson would receive more than $3 million to play next season.
Peterson, who turned 32 last month, left the Minnesota Vikings as one of the best players in franchise history. He was released on March 9 after the team declined the 2017 option on his contract.
The four-time All-Pro had seven 1,000-yard rushing seasons in nine years, including the second-best total in NFL history with 2,097 yards in 2012, but battled injuries the last few seasons.
Peterson also led the NFL in rushing in 2015 with 1,485 yards, along with 11 touchdowns. He has 11,747 career rushing yards with 97 touchdowns in 10 seasons -- 16th on the NFL's all-time list. He is 565 yards behind Jim Brown for 10th place.
The former Oklahoma standout and No. 7 overall pick of the Vikings in the 2007 draft missed all but three games in 2016 after undergoing right knee surgery in September. Peterson totaled 72 yards on 37 carries.
Peterson would have made $11.75 million in base salary with a $6- million roster bonus had his option been picked up by the Vikings.
The Saints already have a 1,000-yard rusher, Mark Ingram, who totaled a career-high 1,043 yards last season. Ingram, 27, has battled injuries since he was drafted in the first round in 2011, with last season marking only the second time he's appeared in all 16 games.