Sources told the NFL Network on Thursday that Cook is close to signing a deal. Terms of the deal weren't disclosed. The team hasn't confirmed the signing.
Cook previously visited with the Saints on March 14, the New Orleans Advocate reported.
The 31-year-old tight end served as quarterback Derek Carr's top pass-catching target last season in Oakland. He recorded 68 receptions for 896 yards and six touchdowns in 2018.
New Orleans has searched for a reliable tight end since trading Jimmy Graham to the Seattle Seahawks in 2015. The Saints have relied on Josh Hill, Coby Fleener and Benjamin Watson at the position in recent years.
Watson announced his retirement from the NFL after last season, leaving the team with a void at the position.
The Tennessee Titans drafted Cook in the third round of the 2009 NFL Draft. He has 425 catches for 5,464 yards and 25 touchdowns in 10 seasons between the then-St. Louis Rams, Green Bay Packers, Raiders and Titans.