Graham is entering the final season of his rookie contract, and Saints general manager Mickey Loomis acknowledged the team has discussed a contract extension.
"He knows we want to have him here long-term, for his entire career, and I think he wants to be here for that same period," Loomis told reporters Thursday, via The Times-Picayune. "These things have a way of working out for us."
Said Loomis: "We've got some guys that are obviously in the last year of their contract, important players to our team. We've talked about it, and some of those may or may not happen. We're just going to see how this thing unfolds over the next couple of weeks. And it's possible (getting a deal done), is really what I'm getting to."
After restructuring deals for defenders Jonathan Vilma and Will Smith, the Saints roughly have $5 million in salary-cap room. Safety Malcolm Jenkins is one of those "guys" entering a contract year, but Graham has to be the priority here.
Graham has the potential to return to his monster form of 2011, when he accounted for 99 catches, 1,310 yards and 11 scores. Wrist and ankle problems limited him last season, but with coach Sean Payton back in the fold, Graham has a chance to become the league's most productive tight end in 2013.
One way or another, the man will be paid.
UPDATE: Graham addressed his contract situation Friday at Saints camp. "Whenever that's gonna happen, it's gonna happen," he said. "I'm not too concerned about it. I know that if I do what I gotta do on the field and I keep being productive and helping this team win, I'm going to be taken care of."
Follow Marc Sessler on Twitter @MarcSesslerNFL.