According to Yahoo Sports, a one-year contract is close to being finalized.
Winston was displaced in Tampa Bay when the Buccaneers signed Tom Brady last month.
By doing a one-year deal with the Saints, Winston would get to learn from coach Sean Payton, quarterback Drew Brees and others. He would fill the role that Teddy Bridgewater did for the team last season before he left for Carolina. With no other starting jobs available, it could be an ideal landing spot.
Winston would compete for the No. 2 job with Taysom Hill, who re-signed with the Saints on Sunday on a two-year, $21 million contract, including $16 million guaranteed, a source told ESPN. Some see Hill as the heir apparent to Brees, and though he has been a standout in a multipurpose role, he isn't yet an established NFL quarterback.
Payton said earlier this offseason that Hill has "earned this opportunity to be our 2," but he has made it clear that the Saints want to have three QBs active and ready to play on game days so that they can continue to use Hill in his versatile role as a runner and receiver. Winston gives the Saints that experience level while also giving them another young quarterback to evaluate as a possible successor to the 41-year-old Brees.
If the Saints wait until 4 p.m. Monday or later to officially sign Winston, he won't count against this year's compensatory draft choice formula.
The Saints will have to get creative, as usual, to make room for Winston financially, since they have only about $4 million in salary-cap space and still need to make room for their draft picks and a possible long-term extension with running back Alvin Kamara, among others. But Winston's deal would likely include voidable years that spread out any cap hit. And as a bonus incentive for signing Winston, if he were to leave in free agency next year and wind up starting for another team, the Saints could possibly earn a high-end compensatory draft pick.
In 2019, his first season with coach Bruce Arians, Winston had 5,109 passing yards for the Buccaneers, but he also had a career-high 30 interceptions as he became the first quarterback in NFL history to throw for 30 touchdowns and 30 interceptions in the same season.
Turnovers have been Winston's biggest issue as a pro, as opponents scored 112 points off his turnovers last season -- the most in the NFL. Overall, he has lost 111 turnovers in his career, the most of any player in the league since 2015. His 19,737 passing yards since 2015, however, are the seventh most of any quarterback in the league.
After the season, Winston made steps to correct his nearsightedness, undergoing LASIK surgery.
The Bucs could have reached a deal with Winston at any point after the season, but Arians was adamant about seeing "what's behind Door No. 2" when it came to which free agents would be available.
Winston, 26, has gone 28-42 in his five seasons as a starter with the Bucs, completing 61.3% of his passes with 121 touchdowns and 88 interceptions. He has developed a strong connection with Pro Bowl receivers Mike Evans and Chris Godwin.
Arians was happy with Winston's progression on the deep ball, as the 2013 Heisman Trophy winner completed 40.2% of his passes of 20 or more air yards in 2019, which was helped by his chemistry with receiver Breshad Perriman. Before last season, he had completed just 30.5% of such passes since being selected with the No. 1 overall draft pick in 2015.
Winston was suspended for the first three games of the 2018 season after an eight-month NFL investigation concluded that he engaged in detrimental conduct in March 2016 by touching a female Uber driver "in an inappropriate and sexual manner without her consent and that disciplinary action was necessary and appropriate." A settlement was reached in a separate lawsuit.
ESPN's Mike Triplett and Jenna Laine contributed to this report.