The 28-year-old, who signed a new four-year contract with the Welsh club last month after winning the league title during his loan spell at Leicester City last season, will undergo a surgery on Monday to remove extra bone growth in his ankle.
"I am bitterly disappointed. There is not a lot I can do about it, but I am determined to work even harder after my operation to play my part for the team on my return," Dyer told the club website.
The winger, who has scored 31 goals since joining Swansea in 2009, never started in Leicester's title-winning league season, but scored once in 12 appearances from the bench.
Swansea, who have four points from four games this league season, travel to Southampton on Sunday.