Sigurdsson returns to Swansea where he enjoyed a successful loan spell in 2012, scoring seven goals in 18 top-flight appearances.
Davies, 21, has signed a five-year contract at White Hart Lane while Vorm, 30, has penned a four-year deal.
Sigurdsson joined Spurs from Hoffenheim in July 2012 but struggled to establish himself in the starting side.
The Iceland midfielder has signed a four-year contract and says he wants to help Swansea back into Europe next season.
"The main aim for the club is obviously to stay in the Premier League but we want to push on even further and get the club back into the Europa League like they were last season," Sigurdsson told Sky Sports News.
"Hopefully we can finish in the top half of the table."
Sigurdsson continued: "I really enjoyed my time here last time.
"I know the club, I know the manager, I know the players and I obviously know the fans and the city as well so that wasn't that difficult for me.
"Leaving Tottenham is because I enjoyed my time there as well for two seasons but Swansea were interested and that's the club I want to play for."
Crystal Palace were reported to have had a bid accepted for Sigurdsson but the playmaker insists Swansea's style of football persuaded him to choose the Welsh club.
"The way they play football was a big factor," he said.
"They've got some great players here.
"Hopefully we'll be a lot better than we were last season and get back to the top half of the table where Swansea belong."
Sigurdsson is Swansea's fifth signing of the summer after Bafetimbi Gomis, Marvin Emnes, Lukasz Fabianski and Stephen Kingsley already joined Garry Monk's squad.
For Tottenham, Davies and Vorm are the club's only arrivals of the window so far and Mauricio Pochettino's first acquisitions since taking charge.
Davies earned a professional contract with Swansea in 2011 and has impressed in the first team, winning the club's Young Player of the Year award for the past two seasons.
Tottenham had to fight off competition from Liverpool to land the young full-back, who has also earned 10 caps for Wales.
Davies' arrival brings into question the future of Danny Rose, who said at the start of last season he would not be happy sitting on the bench at White Hart Lane.
Dutch goalkeeper Vorm also faces a battle for first-team football under Pochettino, with Hugo Lloris already well-established between the sticks at White Hart Lane.
Lloris was one of Tottenham's star performers last term and the France captain signed a new five-year contract at the club earlier this month.
Vorm, who is likely to be used more in cup competitions, moved to Swansea in August 2011.
He enjoyed an impressive debut campaign, winning the Supporters' Player of the Year and Players' Player of the Year awards.
Vorm made 97 appearances at the Liberty Stadium and was also part of Holland's World Cup squad in Brazil this summer.
Spurs confirm the deal
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