Robson, 54, underwent an operation to remove a tumour from his throat at a hospital in Bangkok, where he is currently working as coach to Thailand's national team, United said in a statement.
The club said specialists who had treated the former Manchester United midfield icon were optimistic he would be able to make a full recovery.
"Bryan Robson underwent an operation to remove a tumour from his throat on March 3, 2011 in a hospital in Bangkok," a club statement read.
"The laboratory report revealed a cancerous tumour, which will need further treatment. The oncologist is confident that the condition is treatable.
"Bryan and his family have requested that he can continue with his treatment in private and ask that their privacy is respected."
Robson won 90 international caps during his career, including 65 as captain, despite a string of major injuries. England manager Sir Bobby Robson nicknamed his namesake 'Captain Marvel.'
Robson also enjoyed a successful career with Manchester United, winning three FA Cups, a European Cup Winners Cup and two Premier League titles by the time he left Old Trafford in 1994.
Bryan Robson in cancer battle: Man Utd
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