According to Matt Law of the Daily Telegraph, the Blues want to get the Belgium international tied down to a new long-term deal, with his current terms set to run out in 2020. Real Madrid are said to have held an interest in signing the former Lille star.
"Hazard knows that Chelsea want to reaffirm his position as the club's highest earner with a huge pay rise and there is encouragement from both sides that the forward will sign an extension," wrote Law. He added that the Blues have told Hazard they believe he is capable of winning the Ballon d'Or with them.
Chelsea have reportedly advised Hazard that if he is to do so he needs to improve his performances in the biggest games. And ahead of the team's meeting with Atletico Madrid in the UEFA Champions League, manager Antonio Conte stressed how vital these huge European occasions are for Hazard.
"I think this competition is very important for him, but I think it's important for the team, too," he said. "This competition brings you to another level. I think Eden has all the possibilities to do this."
Hazard was central to Chelsea winning the Premier League in 2016-17, as he rediscovered his form in spectacular style under Conte. As we can see courtesy of Sky Sports Statto, in the key domestic matches he stood up for the Blues:
This season, his preparations for the new campaign were hampered by an injury layoff and Conte has slowly integrated him back into the first-team setup.
Based on the words of encouragement he's been offered by his manager ahead of the meeting with Atletico on Wednesday, he'll be desperate to start and make a big impact against one of Europe's best outfits.
As noted by ESPN FC's Liam Twomey, it appears as though Conte thinks Hazard is ready to be included in the XI again:
Now 26, Hazard is at the point in his career where plenty will expect him to kick on, especially if he is to reach the summit of the game.
There's no doubt he has the natural talent to do so. Last season and in 2014-15, seasons in which Chelsea won the Premier League title, he was sensational in the final third. Hazard can score goals, create chances and leave opposition players flummoxed with his dribbling ability.
Chelsea evidently feel confident he can make that step based on the remarkable salary they're set to offer him. All eyes will be on Hazard, in the Champions League in particular, to see if he has what is takes to become one of the elite players on the planet.