The deal with Sporting for the highly rated 18-year-old, who impressed at the Under-20 World Cup this summer, could rise to £11m depending on performances
The 18-year-old was one of the shining stars at this summer's Under-20 World Cup, helping Portugal reach the round of 16.
Galatasaray announced on Monday that they were keen to bring the talented attacker from Sporting - the same club which Cristiano Ronaldo spent his youthful years at - to the Turk Telekom Arena.
The deal with the Portuguese club that the Turkish giants have made could see the fee potentially rising to €13m (£11m) depending on performances, with Sporting to receive 25 per cent of any future sale figure.
Bruma could feature in the Champions League for Galatasaray this season, where they will face Real Madrid, Juventus and Copenhagen in Group B.