The 34-year-old will join Spurs on a two-year contract on July 1 following the expiration of his Saints deal.
He is expected to provide back-up for Hugo Lloris in north London and most likely to play in the domestic cup competitions.
“This is a fantastic opportunity for me,” he told Tottenham’s website. “When a team like this comes along, it’s impossible to say no. I couldn’t get here fast enough to get it done.
“Everything about the place is world-class. The stadium is probably the best I’ve ever been to – ask any player in the Premier League, and that’s what they’ll say.
“The training ground is unbelievable, to be able to come here and work, and work hard, I’m very privileged. It’s an opportunity I’m very grateful for.”
Forster made 162 appearances over an eight-year spell at St Mary’s, including 24 outings in all competitions last season.
He has been capped six times by his country, with his most recent call-up coming in March.
Forster travelled to the 2014 World Cup in Brazil and was also part of the Three Lions squad for Euro 2016.
Having begun his career with Newcastle, he spent time on loan at Stockport, Bristol Rovers and Norwich before two successful season-long stays with Celtic resulted in a permanent move to the Scottish club in 2012.
He joined Southampton two years later and, after losing his starting role on the south coast, returned to Celtic Park for the 2019-20 campaign before later returning to favour under Saints boss Ralph Hasenhuttl.
Tottenham finished fourth last season under Antonio Conte to secure a Champions League return following a two-year absence.
“It will be a fantastic season, fantastic to have Champions League football,” continued Forster.
“Having achieved that, I’m sure the club will want to go as far as they can in the competition.
“I’ve played almost 50 European games, so if I can come in and help in any way possible, that’s what I’ll be looking to do.”