Swansea boss Michael Laudrup had feared the worst when Dyer fell awkwardly after a challenge from Norwich defender Sebastien Bassong in the first half of the match at Carrow Road in which the winger had opened the scoring after 12 minutes.
Dyer needed oxygen supply before being taken off the field on a stretcher with a leg brace, but the club posted early positive news about the 26-year-old.
"Initial x-ray shows no obvious break to Dyer's ankle. However, he will have further scans tomorrow before final diagnosis," read a message posted on Swansea's Twitter feed on Sunday.
Laudrup said the injury to his diminutive attacker had effected his team, with Gary Hooper firing a brilliant volleyed equaliser just before halftime for the hosts.
"I think the game changed in two ways with the injury," the Dane was quoted as saying by British media. "Everybody could see it was not good when you get carried out like that.
"We are all human, the team had 10 minutes where they suffered a bit and tactically we also lost our main threat with the runs from the second line, like when he had scored the goal.
"Even though it was a great goal to equalise just before halftime, then it is difficult to come back to the dressing room, even in the first 10 minutes of the second half, we suffered a bit, especially at set-pieces.
"In the end 1-1 I think at the end of a very difficult run of games, we can look at the positive things, even if it is a bit difficult with the injury today."
The point takes 10th placed Swansea to 20 after 16 matches of a packed schedule with Laudrup looking forward to the draw for the last 32 of the Europa League later on Monday.
Such continental treats remain a long way off for his Norwich counterpart Chris Hughton, who was disappointed not to have claimed a fourth win in six league matches which would have taken them past Swansea in the table.
The point, though, leaves them in 14th with 18 points, five ahead of the relegation zone.
"At the moment I am disappointed we did not get the win, because we know what it would have meant to the league table," Hughton said.
"But it is so tight and there are a very large group of teams outside of that top seven or eight can beat each other, and I don't see that changing.
"We are just pleased that at this moment we are on a decent run and have been able to pick up a few points, but we have to keep that going."
The former Newcastle United manager was delighted by Hooper's finish, his fourth since joining from Scottish champions Celtic in July.
"It was a fantastic and very special goal," the former Ireland fullback said
"Gary is a finisher, technically a good striker of the ball so always gives himself a chance."