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Sky Sports News has been told Mudryk was on the way to west London for a medical on Saturday.
It is thought Chelsea are confident of securing the transfer and have offered the 22-year-old a seven-and-a-half-year contract.
A statement from Shakhtar on Saturday night confirmed a deal was 'very close'.
Arsenal remained in talks as recently as Friday but were prepared to walk away from the transfer if the deal was not right.
In social media posts, Mudryk had previously indicated Premier League leaders Arsenal were his preferred destination, with personal terms not expected to have been a problem.
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He liked a parody Instagram post depicting him being held prisoner by Shakhtar Donetsk as they continued to hold out for bigger offers, and uploaded posts showing him watching Arsenal games and describing Mikel Arteta as a "top coach".
But he now appears set to join Graham Potter's struggling Chelsea side who have managed just one win in their last five matches.
Chelsea owner Todd Boehly has already sanctioned moves for Benoit Badiashile, David Fofana and Andrey Santos in the January window, while adding the loan signing of Joao Felix from Atletico Madrid.
Why Mudryk is in demand
At the start of last season, he was an unknown outside of Ukraine with a transfer value of under £1m. Now he is one of the most coveted young players in Europe. Following an explosive rise, Mudryk looks destined for big things.
The 21-year-old, who showcased his thrilling speed and one-on-one ability in the Champions League group stage earlier this season, has been described by Darijo Srna, Shakhtar Donetsk's director of football, as "the best player in Europe in his position" after Kylian Mbappe and Vinicius Junior.
There may be a hint of exaggeration to that statement but Srna is not the only one to hold Mudryk in high regard. His list of suitors is a long one with Chelsea closing in on the race to sign him.