The Dutch defender continues to run down his deal at Stadio Olimpico, with free agency beckoning in the summer.
With no fresh terms agreed as yet, the 25-year-old centre-half will soon be cleared to discuss pre-contract agreements with any interested parties.
Lazio director of sport Igli Tare admits there are many of them, with several offers being lined up, but he remains confident that De Vrij will ultimately commit his future to his current club in Serie A.
“We are on the right track and have to decide together,” Tare told TGCom.
“I hope his journey will continue with us. It is something he wants too, because he is very fond of this place and the club. He has many offers from top level clubs, obviously, but his heart beats for Lazio and I hope in the end he’ll stay.
“It could be sorted out by tomorrow, as it’s not an issue of when, but what the deal will be.”
Reports in Italy suggest that De Vrij has already verbally agreed a two-year contract extension with Lazio.
Getting that deal over the line is considered to be a priority, with there unlikely to be more movement in Rome during the winter window.
Martin Caceres has already been snapped up from Hellas Verona and Tare says focus is being locked on those already at the club.
He added: “With the arrival of Caceres, the Lazio squad is more than complete.
“Luis Alberto has a long contract, but we’ll soon meet with his agents to reconsider it, because he has proved to be an important player.”
Lazio currently sit fourth in the Serie A table, 11 points adrift of leaders Napoli with a game in hand.