It was initially feared that he may have broken his ankle as he landed awkwardly after an aerial challenge with Sebastien Bassong.
However, a series of scans on Monday revealed the true extent of the damage.
A club statement read: "The news is positive after the club feared the worst when Dyer was transferred 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."
Swansea remain 10th in the Premier League after Sunday's game as they prepare to entertain Everton this weekend.