Amorebieta is out of contract at the end of the season and he has failed to agree terms with Bilbao over an extension at San Mames.
Bilbao have now withdrawn the offer and the 27-year-old is keen to play in the Premier League after also attracting interest from Russia.
The Venezuela international's representative Gines Carvajal says there were discussions with big-spending Anzhi Makhachkala, but that the centre-half could join Arsenal on a free transfer.
"We have had some contact with Anzhi, they were ready to make a good offer, but my client wants to play in England so the transfer to Russia was not possible," Carvajal is quoted in the Daily Mirror as having told a Russian newspaper.
"We are in talks with Arsenal, he could join them in June."
Amorebieta has made over 250 appearances for Bilbao and has previously been linked with the likes of Everton, Liverpool, AC Milan and Barcelona.