The 27-year-old stated recently that it would be a 'dream' to move to the Santiago Bernabeu and has repeatedly refused to rule out an end-of-season switch to Spain.
Chelsea have offered Hazard a five-year deal to stay at Stamford Bridge, a contract that would see him become the club's highest earner on upwards of £300,000 per week.
The signing of such a renewal would see Hazard surpass the wages of N'Golo Kante and Cesar Azpilicueta, who both recently renewed their terms with Chelsea.
With the star winger's contract due to expire in 2020, Sarri has spoken of the continued commitment to ensuring Hazard remains in a Blue shirt.
“I want Eden here, of course, but I only want him here if he wants to stay here,” the Italian told reporters.
“I know that there are discussions between the club and Eden's agent every week, so if something new comes up the club will call me immediately.”
Alongside wanting to tie down Hazard, Sarri also spoke recently of his wish to hold on to David Luiz and Cesc Fabregas, but with the club having a policy of only offering one-year extensions to players over 30, it could be tricky for the Blues.
Chelsea also want to tie down Antonio Rudiger and Callum Hudson-Odoi to new deals.
Meanwhile, Alvaro Morata continues to blow hot and cold in the English top flight, a recurring theme since his arrival 18 months ago. And Sarri, no stranger to managing strikers who flourish when confident, says the Spaniard reminds him of former charge Riccardo Saponara.
“It's difficult,” Sarri said. “At Empoli I had a wonderful player, Riccardo Saponara, who we sold to AC Milan. Saponara was one of the best players I have ever seen, but a little bit mentally fragile.
“Alvaro has scored four or five goals in his last six matches, so I cannot see a big problem. I hope that there will be more this season.
“Every season he scores 15 or 16 goals, so he's around his average [goal return]. I speak with him every week, but on the pitch he is alone.
“I think that at the moment he's not playing to his full potential. It's difficult.”
Contrary to the comments made by the majority of other Premier League bosses regarding a busy winter schedule, Sarri sees the packed period as an opportunity rather than a problem – though he concedes he will not have time to walk his dog.
“Oh, I have a friend here from Florence [who is staying with me], so I have no problems with my dog!
“It's a difficult period. We have seven Premier League matches but we should see this period as an opportunity, not as a problem. It's a chance to gain points.
“When I was in Serie B in Italy we played three matches – 22, 27, 31 December I think – and it was fantastic. The atmosphere in the stadiums was really good.
“I like very much to play over that period. Of course, it's difficult because we have to play 22, 26, 30, January 2 and 5, so not so easy. But I'm looking forward to it.”