The winger was stretched off during the Welsh side's 1-1 draw with Norwich City on Sunday after a challenge from Canaries defender Sebastien Bassong.
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"Good news: No break for Nathan Dyer. Ankle ligament damage instead though. Can't confirm yet how many weeks out," Swansea wrote on their Twitter account.
In a statement on their official website on Monday, the club later added: "In-depth scans confirmed that Dyer had escaped a break at the expense of a ruptured lateral ligament.
"Although there is also further damage to two other ligaments, the news is positive after the club feared the worst when Dyer was transferred by ambulance to hospital at half-time in severe pain.
"Dyer will see another specialist in the next few days after the swelling has gone down before confirming how long he'll be ruled out for."
The winger has featured 14 times for the Swans so far this season, scoring four times – including giving his side an early lead against Norwich.