21 July 2016 Last updated at 20:58 BST
Speaking in an interview with BBC Newsnight's Evan Davis, he said it was "a job that doesn't exist" which would require a "constitutional change" - and he compared the role to that of "director of football".
Watch the full interview on BBC Newsnight on 21 July at 22:30 BST on BBC Two.
Labour leader Jeremy Corbyn has dismissed an offer from Owen Smith of becoming party president in the event his rival wins the Labour leadership.