Stamkos told the St. Petersburg Times he believes his agents will be meeting with Lightning General Manager Steve Yzerman at this weekend's Entry Draft in Minnesota and that agreement on a new deal could be forthcoming.
"I think it's close," Stamkos told the Times. "I don't think it's far off. As close as we are, it's not something I'm really worried about."
Since being taken with the first pick in the 2008 Entry Draft, Stamkos has recorded a 51-goal season, consecutive seasons of 90-plus points and helped lead the Lightning to Game 7 of the Eastern Conference Finals this past spring.
As a restricted free agent, another team could sign Stamkos to an offer sheet which the Lightning would then have the right to match.
The Times reported Stamkos could earn in the $7 million range per season for his new contract.
"Tampa is a great organization," Stamkos told the paper. "With the success we had this year, there are only going to be bigger and better things.
"You want to be part of that. Whatever happens happens, but I have confidence things are going to get done well before (July 1)."
Stamkos has 119 goals and 232 points in 243 regular-season games with the Lightning. He added 6 goals and 13 points in 18 games during his first trip to the Stanley Cup Playoffs this season.