Left-back Davies, 21, has signed a five-year deal at Spurs while Netherlands international Vorm has joined on a four-year contract to compete with France No.1 Hugo Lloris.
Sigurdsson, 24, returns to the club where he spent a year on-loan before joining Tottenham in 2012, signing a four-year contract at the Liberty Stadium.
Davies said: “Moving to Spurs is a massive footballing challenge for me again, but I’m looking forward to it.
“They were the ones chasing me from the beginning and I want to repay the faith they have shown in me by doing well for them and continuing to improve as a player.’’
The two Premier League clubs have been locked in extensive negotiations over this complex trio of transfers for some time, with Sky Sports first reporting Spurs had opened talks with Davies several weeks ago.
Swansea believed they could attract Sigurdsson despite Crystal Palace having already agreed an £8million fee to take him to Selhurst Park.
And the fact both clubs are currently on simultaneous pre-season tours of the USA has worked in their favour, entering talks and moving the transfer on quickly over the last few days.
Sigurdsson was withdrawn from Tottenham's friendly against Seattle Sounders on Saturday and returned to the UK on Wednesday in order to undergo a medical in the Welsh city.
Vorm, meanwhile, entered talks with Spurs after the clubs agreed a fee over the weekend and underwent a medical in Europe during his extended break following involvement in the World Cup.
The 30-year-old was initially sceptical of his chances of competing with Lloris, who recently signed a five-year deal, but has been convinced of Tottenham's enthusiasm to have two top-class goalkeepers to choose from.
Davies, meanwhile, has been keen on a move to Tottenham ever since their interest was expressed over a month ago and has already trained with his new side in Toronto.