Spacey, who started as artistic director of the Old Vic in 2004, is aiming to raise a £20million fund "to make the theatre fit for the 21st century" before he leaves.
Branagh has had experience of running a theatre company, having co-founded the Renaissance Theatre Company in 1987 with David Parfitt. He is also among the names to have been mentioned in connection as a successor to Nicholas Hytner at the National.
The Old Vic board have appointed Sue Storr, a theatre consultant who also oversaw the appointment of the Royal Court's Vicky Featherstone, to lead the "transparent succession process" to find Spacey's replacement.
A spokesperson for the Old Vic told the Mail's Baz Bamigboye that "the process is only now formally beginning" and there would be "much speculation about names".