Graham was chosen following a "rigorous selection process" validated by the Office of the Commissioner for Public Appointments.
Justice secretary Jack Straw is inviting the justice select committee to hold a pre-appointment hearing today and to report on Graham's suitability for the post.
This is in line with proposals announced last year to increase democratic scrutiny of key public appointments.
Graham has been director general of the ASA since 2000. He is also a non-executive lay representative on the Bar Standards Board, since 2006, and non-executive director of Electoral Reform Services, since 2001.
Richard Thomas has been the information commissioner since November 2002, when he became the second person to hold the job.
Prior to this role he held positions as director of public policy at Clifford Chance, director of consumer affairs at the Office of Fair Trading and head of public affairs and legal officer at the National Consumer Council.
The vacancy for director general of the ASA will be advertised at the end of February.
Graham said: "Director general of the ASA is one of the best jobs in the world, but I've been doing it since April 2000. It's time for somebody else to take things forward to the next level.
"Meanwhile, the information commissioner is so much at the centre of debates on information security, privacy, better government and the right to know that I am keen to take on this new challenge."