Vazquez and his agent are taking a few days to review the proposal and have yet to accept or reject it.
Several European sides have explored the possibility of signing 29-year-old Vazquez, but no concrete bids have been made.
Vazquez is a fringe player for Zinedine Zidane’s side and started only eight of his 18 La Liga appearances last season.
Madrid have given Vazquez the freedom to decide his future and will not stand in his way if he wants to leave.
Other than a loan spell at Espanyol in his younger days, Vazquez has never played professionally for anyone but Madrid.
The Galician switched agents last season to Arturo Canales, who also represents Gerard Pique, Ander Herrera, Ivan Rakitic and others.
Were Vazquez to accept a move to Qatar, he wouldn’t be the first Spaniard to walk that path.
Santi Cazorla joined Xavi Fernandez’s Al Sadd this summer, while former Madrid player Alvaro Mejia plays for Al-Shahaniya.