Officials from the Blues have travelled to Poland, where Shakhtar are based, and talks are ongoing.
Shakhtar want £85million for Mudryk and have maintained throughout this transfer window they will not sell for less.
Arsenal have made two bids for their no1 January transfer target and have indicated over the past few days they are now willing to part with £60m upfront.
Talks are advanced with the Gunners and negotiations with them are centred around the structuring of add-ons.
Mudryk has so far stated a preference to join Arsenal and that is why the Gunners are considered favourites to win the race for him.
Chelsea coming in, however, complicates matters and they have shown a willingness to get closer to the £85m that Shakhtar want.
The Ukrainian club want a better quality of add-ons from both Arsenal and Chelsea though, according to sources with knowledge of the deal.