The fee is believed to be around £13m.
Allen had one year left on his contract at Anfield but was believed to have fallen out of manager Jurgen Klopp’s plans.
He is the second of Mark Hughes’s close-season signings after Egypt’s Ramadan Sobhi.
“It’s been a whirlwind 48 hours for me but I am delighted to have finally been able to put pen to paper,” Allen told the club website. “I was really excited as soon as I heard of the club’s interest and it wasn’t a difficult decision for me to make.
“I am incredibly excited about my future here at Stoke City.”
‘It’s relatively out the blue, I’d seen that the offer was accepted and things have moved quickly since then so glad to get it done quickly and I can’t wait to get out and meet the lads in Florida and get the football underway.’
Allen will fly out to link up with his new team-mates in Orlando on Tuesday after becoming manager Mark Hughes’ second summer signing, a few hours after Egypt winger Ramadan Sobhi arrived from Al Ahly.