The Gunners have made Mudryk their top target in this transfer window and have already had an initial bid of around £55million rejected.
Talks have continued between the two club since then and it is understood Arsenal have now conveyed a willingness to increase their offer.
Mudryk has been scouted by several clubs over the past 18 months but it is understood that Shakhtar are currently only in talks with Arsenal.
The first bid of £55m from Arsenal, which included around £20m in add-ons, was made as a way to kickstart talks.
Regular dialogue has now opened up between the two clubs and there is confidence an agreement can be found for Mudryk to secure the move he desires.
Shakhtar have publicly stated they value the 21-year-old at around £85m, however Arsenal hope a deal can be done for less.
It is thought the structuring of add-ons and the up-front fee will be important to finding an agreement that suits all parties.
Mykhaylo Mudryk - Arsenal target
Mudryk has expressed a strong desire to join Arsenal, posting on social media during the last two Gunners’ matches to show he has been watching.
Arsenal’s priority for the January transfer window has been to add a wide forward and Mudryk is their No1 target.
Mikel Arteta’s side are currently seven points clear at the top of the Premier League and Arsenal want to add to their squad to bolster their title charge.