The 28-year-old had increasingly become a fringe figure in the Spanish capital last season, making just 18 in La Liga – of which just eight were in the starting line-up.
The adaptable Vazquez has won three successive Champions League titles and two La Liga crowns with Madrid since returning to the club from Espanyol in the summer of 2015.
Indeed, he has clocked up 206 first-team appearances across that spell but is finding himself increasingly surplus to requirements and Madrid will allow him to leave the club this summer if that is his desire.
Tottenham and Napoli were two of the teams that were linked with a summer transfer for the player but he now appears more likely to join the unnamed Qatari club should he leave the Spanish capital.
Vazquez’s contract at Madrid expires in June next year and he could follow Santi Cazorla in swapping La Liga for Qatari football this summer.