The 29-year-old limped off after just seven minutes of the 2-0 win over Leicester City on Sunday.
A Premier League winner while on loan at Leicester last season, Dyer will have surgery in due course.
The former Southampton player has featured in five games since Paul Clement was appointed Swans manager last month.
"The initial prognosis is it doesn't look good," Clement said.
Swansea are also without fellow wingers Jefferson Montero (torn hamstring) and Mo Barrow, who is on loan at Leeds United.

A masked man holding a gun approached him at about 10:35 BST in a car park in Glenarm on Sunday and demanded the man's car keys.
He then made off from the car park on New Road in a red Ford Fiesta.
A robbery at a shop on Coast Road in Ballygally in County Antrim was reported about 15 minutes later.
A masked man with a gun entered the premises and left with a sum of cash after demanding money from staff.
The red Ford Fiesta was found abandoned at about 11:15 BST on Dickeystown Road in Glenarm.
Police believe the two robberies could be linked.
No-one was injured in the incidents but both the man whose car was stolen and the shop staff were left "very badly shaken", police said.
Police are appealing for anyone who saw the car to contact them.

On Friday, the Florida jury ruled Gawker had violated Hogan's right to privacy, awarding him $115m (Â£79m) for economic damage and emotional distress.
But jurors decided on the additional measure as a punishment for the company and a deterrent to others.
Gawker, who warned the verdict was already "debilitating", will appeal.
The video was posted in 2012 after Hogan, whose given name is Terry Bollea, was secretly recorded having sex with his friend's wife.
Hogan's legal team successfully argued the New York-based website violated his privacy and the video was not newsworthy.
The closely-watched case raised questions about the freedom of the press in the digital age and could lead to greater caution among online sites.
Gawker Media was hit with a $15m ruling, while owner Nick Denton was stung with an additional $10 million.
The editor who posted the video and wrote the accompanying post, AJ Daulerio, must also pay $100,000.
However correspondents say the company stands a good chance of getting the damages reduced or reversed on appeal.
The jury was told that Gawker Media is worth $83 million, while Nick Denton has a total of $121 million. Mr Daulerio has no assets, the court heard, only thousands of dollars in student debts.
Swansea City winger Nathan Dyer has been ruled out for the rest of the season with a ruptured Achilles tendon.
[NEXT_CONCEPT]
A man has been robbed at gunpoint as he was taking photographs in a village on the County Antrim coast.
[NEXT_CONCEPT]
Ex-professional wrestler Hulk Hogan has won a further $25m (Â£17m) in punitive damages from gossip site Gawker over the publication of a sex tape.