Alex's transfer to Chelsea was delayed when the Brazil defender flew out of Britain following a request by immigration officials to clarify his status.
The 25-year-old Alex, who was set to join the Blues after two years at PSV Eindhoven, was cleared to enter Britain but chose to return to Brazil voluntarily, according to Chelsea.
British papers reported Tuesday that Alex was irked when officials told him he was unlikely to receive the work permit he would need to play in England.
Players from outside the European Union need to have played in 75 percent of their country's internationals over the most recent two years to automatically receive a permit, although clubs can appeal under extenuating circumstances.
Alex has not been a regular pick for Brazil but was still expected to be a key player for Chelsea next season.
Chelsea struggled last season when captain John Terry was injured, with manager Jose Mourinho pointing to the central defender's absences as one of the main reasons the club failed to win a third straight Premier League title.