The 26-year-old joins the North London outfit after spending most of his career at Leverkusen, where he made 304 appearances.
Various reports in the media claim the fee is around £19.3m for the goalkeeper, who has six caps for the German national side.
Gunners head coach Unai Emery expressed his delight about the German's arrival.
“We are very pleased that Bernd Leno will be joining us. Bernd is a goalkeeper of high quality and experience. He has been a top performer and regular number one goalkeeper with Leverkusen in the Bundesliga for the past seven years. We are all excited that Bernd has chosen Arsenal Football Club and look forward to start working with him in pre-season," Emery told the club's official website.