The Serbo-Montenegrin star will be out of contract with the Roman giants at the end of this season, and Lazio after desperate to tie him to a fresh deal.
There has been no shortage of interest throughout Europe in the 25-year-old, with the likes of Real Madrid, Juventus as well as two unnamed English clubs all enquiring about the player's situation.
However, after talks with Vinicio Fioranelli, Lazio's chief officer Luca Baraldi declared himself cautiously optimistic over the club's prospects of retaining Stankovic.
"I can say the meeting has been positive," Baraldi told RAI television. "We have removed sharp corners that were there in the past.
"We are on the right way, now we will see where it takes us.
"For us Dejan is a very important player. Difficulties are bounded to the fact that he's a player who is looking to secure the most important contract of his career.
"Of course big teams are interested in him; Juventus, Inter, Real Madrid have made us big offers.
"I don't know what will happen. Let's hope there could be good news."
Baraldi went on to confirm that the club had tabled Stankovic a lucrative new five-year contract offer, and insisted whatever happened the player would not be quitting Rome before the end of the season.
"We made an important offer for Lazio and in today's football economic situation, we offered him a five-year contract.
"The agent says there is quite a big difference between our offer and the other ones they have received.
"But we are convinced that this offer is right, especially as it's a sum we can pay.
"I absolutely exclude that the player could leave Lazio before the end of the season."
