The Brazilian-born Croat will complete a deal to join the Ukrainian team in the next 48 hours after being left out of the Gunners' 27-man squad for a pre-season training camp in Austria.
After Bilic gave his blessing to the move, reassuring the striker that plying his trade in the Ukrainian league would not affect his international credentials, the 27-year-old completed a medical at Emirates Stadium on Monday under the supervision of Shakhtar's medical staff.
Bilic said in the Daily Mail: "I spoke with Eduardo. He was very pleased with the move to Shakhtar. He is going to a strong league.
"This season they will once again play in the Champions League, have many difficult matches and this is excellent for the national team."
Shakhtar, winners of the last ever UEFA Cup in 2009, have reportedly agreed to match Eduardo's £2 million-a-year salary at Arsenal and he was convinced to move to the Ukraine following further conversations with Croatian skipper Darijo Srna, who plays for Shakhtar.
"The captain of Shakhtar is Darijo Srna, in the team there are a lot of Brazilians — I have no doubt that Eduardo will quickly adapt to Donetsk," Bilic added.
The forward has struggled to fully recapture the form that earned him a move to the Gunners after suffering a horrific double leg break in a match with Birmingham City in 2008, but Shakhtar coach Mircia Lucescu hopes he can help him regain his confidence.
"Yes, he had a serious injury, we are aware of the risk," Lucescu said in The Sun.
"But at the same time we realise he can give a lot to Shakhtar. We expect a lot from Eduardo.
"We wanted to buy Eduardo before he moved to Arsenal. Not everything went his way last season, but he is still a player of very high quality, a real marksman."
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