The versatile 24-year-old collapsed midway through Boro's Capital One Cup victory over Gillingham earlier this week.
"He will now see a specialist to decide if he needs an operation or not but it's still going to be a considerable time out.
"It's a blow to both Rhys and to the club. We were going to depend on his quality and athleticism this year."
Mowbray has acted swiftly to cover the absence of Williams, and fellow injured defender
Socceroos coach Holger Osieck believes veteran goalkeeper Mark Schwarzer has a key role to play in their FIFA World Cup qualifying campaign.