The defenders are both believed to be on the verge of signing a contract extension with the reigning Spanish champions after successful talks with the club on Tuesday
Quilon arrived at Madrid's headquarters on Tuesday evening, only to leave two hours later after successful negotiations over contract renewals for his clients.
The 29-year-old Arbeloa, whose current deal is due to expire in June 2014, joined Madrid from Liverpool in 2009, and has since developed into an important first-team member. He has made 82 Liga appearances for Los Blancos since his return to the club.
Albiol has been struggling to secure a regular first-team spot at Madrid since joining the club from Valencia three years ago, but recently admitted that does not wish to leave the Spanish capital. The 26-year-old's current contract also runs until the summer of 2014.
Both players have yet to put pen to paper, but are expected to make their renewals official in the coming days.