The centre-back was injured during Hull's 2-0 loss at Premier League leaders Chelsea last weekend, with manager Steve Bruce initially fearing that Dawson would miss six weeks of action.
Dawson, who earlier this campaign missed a month with an ankle problem, is now set to be absent for a hectic end to December as Hull play four matches in the space of 13 days.
Bruce is already without injured duo Robert Snodgrass and Mohamed Diame, while Tom Huddlestone is also suspended.
And, with the likes of Nikica Jelavic and Abel Hernandez having also been absent this term, Bruce is desperate for a change in luck as he looks to steer the club out of the bottom three.
"Unfortunately it's a grade two tear [for Dawson] and it could be anything from three, four, five weeks minimum I'd have thought," said Bruce ahead of Saturday's match with Swansea City at the KC Stadium.
"That's happened too much to us from the first day of the season with Robert Snodgrass to Dawson, to Nikica Jelavic to Mohamed Diame.
"It's been one thing after another in that department.
"We could do with a little more luck because we've had four or five with pretty ugly injuries and we could do without it."
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