The 26-year-old right-back suffered the injury when he was challenged by David Meyler in the 53rd-minute of Saturday???s 3-1 victory over Hull at the Liberty Stadium.
Hull midfielder Meyler was shown a straight red card for the tackle, while Naughton had to be helped from pitch and then taken to hospital.
And the Swansea website reported on Tuesday morning: ???An initial scan on Saturday ruled out a fracture, but a follow up scan last night confirmed ligament damage that will keep the January signing from Spurs out for around six weeks.
???With just seven league games left, it???s a blow for Naughton and the Swans, the right-back having continued to impress with every appearance.???
The Swans paid ??5million to sign Naughton at the start of the year and he has started their last nine Premier League games.
Swans boss Garry Monk said after Saturday’s match that he felt Meyler’s tackle had merited a red card.
“It was a genuine 50-50 ball and my initial reaction??was that it was bad challenge,” Monk said.
“I don’t think there was any malice from the player but you always run that??risk when your foot goes over the ball.”
Hull boss Steve Bruce said: “The referee has given the decision by what he sees and, from his angle he might see it as reckless, but you may as well say in these grey areas you’re not allowed to tackle on the floor.
“We all want our team to fight and compete, because that’s what the Premier??League is.”
Swansea fan favourite Angel Rangel came on to replace Naughton on Saturday, and the Spaniard is likely to continue to deputise over the remainder of the campaign.
Rangel said: ???It???s not good, because Kyle has an injury and I don???t wish bad on anyone.
???But, from my point of view, I always try and be ready so, if the manager needs me, I???ll try my best.???