The trio were among the stand-out performers during Swansea's run to an 11th-place Premier League finish last term, and have started the new campaign strongly.
Taylor, who still has three years left to run on his current deal, will be out for the rest of the season after breaking his ankle in three places in the 2-2 draw with Sunderland.
And the 23-year-old left-back has thanked the club for electing to press ahead with his new deal despite the severity of his injury.
He told the club website: "We've been talking about a new contract for a few months now - and we were very close before the injury happened.
"But in fairness to the club they have already said they will honour the new deal despite the fact that I won't play again this season. The way they have handled it speaks volumes about the club and the board."
Centre-half Williams and goalkeeper Vorm have around 18 months remaining on their contracts, but the south Wales club are keen to tie their key performers down to long-term deals.
The Swans have also opened talks over a new deal with Wales Under-21 keeper David Cornell.
Source: PA
Source: PA