A proposed deal for Clucas to join Burnley broke down earlier this month, with the Potters taking their opportunity and tying the former non-league player to a four-year contract.
"I think this is a great deal for us as a football club, especially when you take into account the sum of money he moved for just a year ago," Stoke manager Gary Rowett told the club's official website.
"Sam boasts real Premier League quality, and the versatility that he offers will be hugely important to us, as he can play in any of the midfield areas, he is very comfortable out wide and he can play at left-back too," he added.
"Most importantly for me though, he has a great character, he is incredibly mobile and he is exceptionally good technically - he gives us a lot of different options and we are delighted to welcome him to the club."
Chief Executive Tony Scholes added: "Sam is undoubtedly a Premier League calibre player and has been one of Gary Rowett's prime targets during the transfer window, so we are delighted to have managed to get the deal over the line."