Former England player Dawson, signed from Tottenham Hotspur in the close-season, suffered a grade two tear during Hull's 2-0 defeat at Chelsea last weekend -- a result that left them in the Premier League's relegation zone.
"It's a grade two tear like we thought so we're looking at three, four or five weeks minimum, which unfortunately has happened to us too much," Bruce told reporters.
"It's been one thing after the other. It's part and parcel of it but we could do with a bit more luck.
"We've got three or four with pretty ugly injuries but we have to get on with it."
Hull host Swansea city this weekend, before facing Sunderland, Leicester City and Everton over Christmas.
Midfielder Tom Huddlestone will also be unavailable after being shown a straight red card against Chelsea.
Huddlestone, another former Tottenham player, begins a four-match ban against Swansea.
(Reporting by Martyn Herman; editing by Toby Davis)
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