General Manager Steve Yzerman has until 12 p.m. ET on Friday to sign Stamkos or else he will become a restricted free agent and, therefore, be open to signing an offer sheet from any other team.
If Stamkos were to sign an offer sheet after 12 p.m. ET Friday, the Lightning would have seven days to match it or risk losing him.
Yzerman has reportedly been in negotiations with Stamkos' camp.
"Nothing new to report, (but) anytime you are talking it's positive," Yzerman told the Tampa Tribune.
"All I can say is that we are still working on it," Mark Guy, the agent for Stamkos, wrote in a text late to the Tribune Thursday night.
While there had been reports stating Philadelphia's interest in sending Stamkos an offer sheet, Flyers GM Paul Holmgren said Thursday it was not going to happen.
Any team that submits an offer sheet to Stamkos should he hit restricted free-agent status must possess enough salary cap space to sign him (they can go over the $64.3 million salary cap by 10 percent until the regular season begins), and must have in their possession enough draft picks that are their own to give as compensation.
In Stamkos' case, the signing team would have to surrender four first-round picks based on the expected average annual salary he would receive.
Stamkos has the most goals in the NHL during the past two seasons (96). He shared the Rocket Richard Trophy with Sidney Crosby in 2009-10 with 51 goals and finished second behind Corey Perry this past season with 45.
Follow Dan Rosen on Twitter at: @drosennhl