The defender, who is suffering from a hamstring injury, has appeared in just two league games all season.
Hudson suffered the injury during the 2-0 defeat against Manchester United in January when he was forced off the pitch after 70 minutes at Old Trafford.
Hudson said on Twitter: "Unfortunately, I am out for the rest of the season. The hamstring injury at Man U was worse than first thought."
Ole Gunnar Solskjaer's team, who are second from bottom of the Premier League table, entertain Hull City on Saturday.