The 29-year-old defender has been in talks with the Rs over a potential transfer across west London to the Loftus Road outfit.
However, it is understood Alex will not be joining Mark Hughes' side as the club are unwilling to match his wage demands.
The Brazil international handed in a transfer request at Chelsea in November and has subsequently been linked with a move to QPR, as well as Juventus and Paris St Germain.
The Hoops were granted permission to speak to Alex but have failed to agree terms, with reports suggesting PSG are in the driving seat for his signature.
The French club's director of football Leonardo confirmed on Tuesday "there is the possibility" of signing the defender, who would link up with former Chelsea manager Carlo Ancelotti should he move to the Parc des Princes.
Meanwhile, QPR have declined to comment on reports that the club are encouraging Anton Ferdinand to shake hands with Chelsea's John Terry before Saturday's FA Cup tie.
The fourth-round tie will see Blues captain Terry face Ferdinand four days ahead of the court case in which he stands accused of racially abusing the Rangers defender.
Extra police have been laid on for what will be a potentially explosive west London derby, the first time the sides have met since October's controversial Barclays Premier League clash.
Television footage emerged after that game which prompted allegations Terry had racially abused Ferdinand, something the Chelsea and England captain denies.
Press Association Sport understands the clubs will issue a joint statement in the next 48 hours in a bid to lower tensions around the match.