Pope, who joins the club for an undisclosed fee, believed to be £12m, on a four-year contract, as he looks to stay in contention for the England squad in World Cup year after the Clarets’ relegation.
He becomes Eddie Howe's second signing of the summer, following Matt Targett, and the 30-year-old said: "Now I'm here, I can't wait to get started.
"The deal has taken a couple of weeks to come to fruition, but it got over the line really quickly and I'm delighted to be here and I'm really looking forward to getting stuck into it."
Head coach, and former Burnley boss, Eddie Howe added: "Nick is an exceptional Premier League and international level goalkeeper, so I’m very pleased to be adding strong competition to a very important position.
“There has been a lot of interest in him from other clubs this summer, so I’d like to thank our owners and everyone behind the scenes for getting us ahead of the competition.
“I’m delighted he is joining us as we prepare for the exciting challenge ahead.”