The Potters have agreed a fee which could rise to six million euros (Â£5million) for the 19-year-old, who has been capped six times by his country.
Winger Ramadan becomes Mark Hughes' first signing of the summer with Stoke having also tabled a Â£17million bid, rising to Â£20million with add-ons, for West Brom's Saido Berahino
Stoke chief executive Tony Scholes told the club's official site: "We are absolutely delighted to have signed a player of Ramadan's undoubted quality and potential.
"He's a special young talent who is excited about the prospect of making an impact in the Premier League and we're looking forward to giving him that platform."
Sobhi has been granted a work permit and is expected to receive his UK visa in the next 48 hours
He should then join his new team-mates in Orlando later this week where they are in pre-season training.
He is a graduate of the Al Ahly youth system and made his professional debut as a 16-year-old, earning his first international cap a year later.
He helped Al Ahly win the Egyptian League title last season under former Tottenham and Fulham manager Martin Jol.
Source : PA