Graham will turn 32 in October but has made a successful return this season from successive concussion blows that ended his 2021 campaign early.
The backrower admitted he entered the season unsure whether he would play beyond this season, with his contract up at the end of this year.
But after returning to the Sharks's starting side as part of their top-four run, Graham is now certain he wants to play on and is confident another deal will materialise.
"I definitely want to play on," Graham said.
"I have had discussions with the club ... we're both comfortable with how they've progressed.
"Obviously it's a bit to work through. There's a lot whole lot of moving pieces at the moment, not just for next year but for the future.
"I want to see the club move forward into the future in good areas. And certainly the squad we have at the moment has those players in there.
"So the priority for me is getting those moving pieces a little more stable and then my stuff can just filter on in the background."
Graham said he was only considering a one-year contract at this point and would assess his future each season thereafter.
Graham's focus is firmly on the chances of a second title as one of the two last remaining members of Cronulla's 2016 squad along with Andrew Fifita.
After an ankle injury initially pushed back his comeback from concussions, Graham has missed just one game since his round-eight return to clear any internal doubts he had about his future.
"I just wanted to get some game time under my belt. It was almost 12 months I spent away from the game," Graham said.
"It's the hardest competition in the world. Just that grind you go through every week and it takes a lot of mental determination to get up for not only the games but training.
"The game is certainly not slowing down and the young guys coming through are just getting better and more professional.
"You need to still know you have that inner belief to compete with these guys every day. Those were a few boxes I had to tick."
Graham has moved into radio work covering football this season and has eyes on coaching at Cronulla while currently doing his accreditation alongside Benji Marshall.
"It's certainly an option, I enjoy that part of the game," he said.
"Obviously there's a lot that goes into it.
"The systems and the gameplay and trying to figure out attacking styles and defensive styles and mentoring young guys."
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