The Seahawks have offered a range of picks that would include at least one future first-rounder.
Fox Sports first reported the trade talks between the teams.
Graham, 28, is regarded as one of the best pass-catching tight ends in the NFL, a man who often lines up as a wide receiver. He has 4,752 receiving yards and 51 touchdowns since entering the NFL as a third-round draft pick in 2010.
In his five seasons with the Saints, Graham had more receptions, receiving yards and receiving touchdowns than any other tight end in team history. According to ESPN's Stats and Information, Seahawks tight ends combined for 48 receptions in the 2014 season, sixth fewest in the NFL. Quarterback Russell Wilson didn't target his tight ends once in the Super Bowl loss to the Patriots. It was the first time in his career he didn't target a tight end in a game.
Saints coach Sean Payton and general manager Mickey Loomis had made it clear they didn't plan to sit idly by after a disappointing 7-9 season in which they underachieved on both sides of the ball -- Graham included.
Graham caught 85 passes for 889 yards and 10 touchdowns last season, but he was less effective than ever as a downfield threat, averaging a career-low 10.5 yards per reception.
Graham battled a significant shoulder injury for most of the season, which affected his ability to dominate physically off the line and in the red zone. He still played in all 16 games and had some success fighting through it with some big performances in the middle of the season. But he fizzled down the stretch, with just 20 catches for 219 yards and one touchdown over the final five games.
Unger, 28, is a two-time Pro Bowl selection out of Oregon. He was one of only three players on the Seahawks that were on the team before Pete Carroll became head coach in 2010, along with nose tackle Brandon Mebane and punter Jon Ryan. Unger played in only six regular-season games last year because of knee and ankle injuries.
The Seahawks released tight end Zach Miller last week, so it was clear they were in the market for another tight end. Graham is starting a four-year, $40 million contract that will pay him $10 million this season and has $20.9 million in guaranteed money.
The Saints also released veteran linebacker and captain Curtis Lofton and longtime running back Pierre Thomas in recent days, with more big moves still possible. Meanwhile, top cornerback Keenan Lewis voiced his displeasure Tuesday over the Saints' unwillingness to guarantee the final three years of his contract -- insisting that they trade or release him if they're not willing to commit to him.
ESPN's Chris Mortensen, Adam Schefter and Terry Blount contributed to this report.