The disclosures come in an interview to be broadcast on the digital channel Teachers' TV.
Not known for his modesty, the member once dubbed "Gorgeous" and former Celebrity Big Brother contestant, said: "I was quite popular at school."
He added that he was already "running wild" when he arrived at Charleston primary, Dundee and was "inordinately proud of being known as the best fighter in school."
Mr Galloway has posted on the Friends Reunited website welcoming messages from old "friends and foes". He said: "If I were to meet someone now that I had punched - or that peculiarly Scottish form of assault, butted - I am not sure what I would say except sorry."
The MP for Bethnal Green and Bow, who was expelled from the Labour party for comments over the Iraq war, reveals that one of his first acts of rebellion was refusing to wear a cap to Harris academy, Dundee. He also remembers an "extraordinary altercation" with the head there: "Mr Hamilton told me that I must not sport a moustache - I was only 15 after all."
"School Days", Wednesday at 10.15pm