The Gunners have made signing a new wide forward a priority in the January transfer window and are prepared to back Mikel Arteta to help maintain their title push.
Premier League leaders Arsenal have been tracking Mudryk since the summer and have now entered negotiations with Shakhtar in the hope a deal can be struck.
Shakhtar value the 21-year-old at £85million but it is thought they could soften their stance. Arsenal wanted to sign a wide forward during the summer and they looked at Pedro Neto of Wolves.
The need for more firepower in attack has increased because striker Gabriel Jesus is facing three months out after knee surgery.
Shakhtar are reluctant to sell Mudryk next month as they are aiming to win the Ukrainian top-flight, which would secure qualification to the Champions League. They want their star winger help them achieve that goal and then assess his options next summer.
Mudryk is attracted to the prospect of playing for Arsenal and publicly revealed his admiration for them during a recent interview. There is, though, currently no indication that he would look to force through a move should Shakhtar dig in their heels and take a resolute stance over his future.
Mudryk is regarded as one of the best young prospects in Europe and has also been scouted by Manchester City. Brentford were interested in signing him last summer and Shakhtar also rejected an offer of £25m from an unnamed Premier League club, believed to be Everton.
Other clubs are expected to join Arsenal in the race to sign him Mudryk, who impressed in the Champions League this season as Shakhtar were knocked out in the group stage. He scored three goals in six games to help Shakhtar finish third in their group, behind Real Madrid and RB Leipzig, to qualify for the Europa League.