Financial terms were not disclosed for the three-time Pro Bowl selection, who will sign the deal on Thursday.
Bryant acknowledged the move by responding to NFL Network insider Ian Rapoport's report with the following tweet: "#ThrowuptheX".
The 30-year-old Bryant, who was released by the Dallas Cowboys in April, had visited other teams in the offseason -- including the Cleveland Browns in August. He joined fellow wideouts Brandon Marshall and Kamar Aiken in working out with the Saints on Tuesday.
Bryant could get a look at his former team on Nov. 29 when New Orleans plays at Dallas.
"Dez is a great player," Cowboys head coach Jason Garrett said Wednesday, per the Dallas Morning News. "Really happy for him and that opportunity. I know he'll help their team.
"I love Dez. I wish nothing but the best for him. That's a great spot for him."
Bryant had a team-leading 69 receptions for 838 yards and six touchdowns with the Cowboys in 2017. The 6-foot-2, 220-pound Bryant has gone a career-long 23 regular-season games without a 100-yard performance.
Pro Bowl wideout Michael Thomas has flourished this season for the Saints (7-1), recording 70 receptions for 880 yards and five touchdowns. The 25-year-old finished with 12 catches, a franchise-best 211 yards and a touchdown in New Orleans' 45-35 win over the Los Angeles Rams on Sunday.
Outside of Thomas, however, the Saints have received lukewarm production from the wide receiver position.
Rookie Tre'Quan Smith has 12 receptions for 214 yards and three touchdowns for the Saints, who saw fellow wideout Ted Ginn Jr. placed on injured reserve last month and Cameron Meredith struggle this season.
New Orleans carries a seven-game winning streak into Sunday's contest at the Cincinnati Bengals (5-3).