The long-serving custodian had a clause in his contract which allowed him to leave the Britannia Stadium on a free transfer if he was offered a chance of a move by another club.
Simonsen, who spent the closing stages of last season on loan at Championship side Sheffield United, leaves after six years at Stoke.
The 31-year-old joined Stoke on a free transfer from Everton but found himself third choice behind Denmark international Thomas Sorensen and Asmir Begovic.
Stoke boss Tony Pulis said: "There was a clause in Simmo's contract which allowed him to move on a free if he received the right offer.
"He did well in his time at Sheffield United when he went out on loan in search of first-team football and that has led to this chance."
Pulis added on the club website: "Simmo came to the club in my first spell as manager and has been a great servant throughout his time here, playing a big part in us winning promotion, so we would like to thank him for his contribution and wish him well."