Real Madrid value the Belgium international at no more than €100m (£88m) due to the fact that he is entering the final 12 months of his contract at Stamford Bridge.
Chelsea are said to be under no pressure to sell and negotiations over a proposed deal are being handled by Blues director Marina Granovskaia.
Hazard revealed earlier this month that he had made a decision on his future and expected it to be resolved following the Europa League final against Arsenal on Wednesday.
Current Chelsea boss Maurizio Sarri, who will also have talks over his future following the game in Baku, insisted the club needed to “respect” the decision Hazard has made about his future.
Hazard, who turned 28 in January, signed for Chelsea from French side Lille in 2012 for £32m and has since scored 108 goals in 351 appearances.
During his seven years with the west London club, Hazard has won the Premier League twice (2015 & 2017) as well as enjoying success in the Europa League (2013), the League Cup (2015) and FA Cup (2018).
Belgium football expert Kristof Terreur revealed to Sky Sports the potential deadline for Real Madrid to get a deal for Eden Hazard done.
“Real are wanting to make the final push after the Europa League final. I think the jigsaw pieces might fall into place in that short period after the Europa League final.
“That’s what the player’s hoping for as well. He wants everything done before he joins Belgium – they’re playing Kazakhstan and Scotland – that’s on the fourth of June.
“So there’s a bit of a deadline, but there’s some space to do the deal.”