Rangers had a bid turned down for the wantaway Brazilian earlier in the week but have now reached a consensus with their west London rivals.
They must now convince the 29-year-old to become Mark Hughes' first signing as manager, amid reports he may no longer wish to remain in England.
Chelsea accepted QPR's offer less than 24 hours after Gary Cahill travelled to London to complete the formalities of his move from Bolton.
Alex, who has also been linked with Paris St Germain and Juventus, was banished from the Blues' first-team squad in November after submitting a transfer request.
His signing would represent another coup for ambitious Rangers after they convinced Hughes to succeed Neil Warnock as manager on Tuesday.
As well as having a previous bid for Alex snubbed - reportedly £3million - they have had offers turned down for Andy Johnson, Christopher Samba, and Steven Pienaar.
Should Rangers complete Alex's signing, he will have to wait to make his debut as the club missed the deadline for Sunday's Barclays Premier League game at Newcastle, while he is ineligible for Tuesday's FA Cup third-round replay against MK Dons.
His first game would therefore be next Saturday's league match against Wigan, which is followed by a possible FA Cup clash with Chelsea.