Stoke have paid an initial £10m, but the fee could rise to £12m dependent on the fulfilment of various clauses.
"These fantastic deals are a statement of this football club's intent," said City's chief executive Tony Scholes.
"The club has made incredible strides forward over the past five years and to attract players of this calibre shows our ambitions as we look to continue that progress this season and beyond that."
Manager Tony Pulis added: "There has been a real buzz of excitement around the city about where this club is going since we reached the FA Cup final and then started our Europa League adventure.
"These signings certainly add to that great sense of anticipation.
"Peter has an excellent goalscoring record at the highest level. His goals-per-games ratio for England is remarkable and he scored seven goals in 10 Champions League games for Tottenham last season, which gives you an idea of his pedigree."
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