Rewards Management Middle East, which operates travel reward programmes in Qatar, United Arab Emirates and Bahrain, is 40% owned by HSBC Holdings.
It was founded in 2000 by LMG, HSBC and Al-Seer, the Dubai-based owners of the local Spinney's supermarket chain. LMG acquired the additional stake in the business from Al-Seer.
LMG was itself acquired by Canadian firm Aeroplan in December for a reported £368m.
LMG said its Air Miles Middle East programme was currently receiving 15,000 new applications a month, and that increasing its stake in Rewards Management Middle East would help extend the reach of its global loyalty programme.
Alex Moorhead, chief executive officer at LMG, said: "LMG looks forward to continuing its strong relationship with HSBC and working more closely with the RMME management team to consolidate its loyalty leadership position in this fast-growing region."