Centre-back Dawson, 32, was injured during the club's pre-season friendly victory over Mansfield on Tuesday.
It means the Tigers have just 13 fit senior players less than a month before the new season starts.
On Monday, full-back Moses Odubajo was ruled out for up to six months with a knee ligament injury.
Earlier this month, goalkeeper Allan McGregor (back) and defender Alex Bruce (Achilles tendon) were ruled out for between four and six months.
Hull begin their Premier League campaign against champions Leicester on Saturday, 13 August (kick-off 12:45 BST).
Hull's injury problems have worsened, with captain Michael Dawson ruled out for three months because of a medial knee ligament injury.