Hope, who has been with the paper since 2003, has been promoted from his former role as home affairs editor. He has held numerous other roles including business correspondent and industry editor.
Whitehead is joining the Daily Telegraph from the Daily Express as home affairs editor. He was home affairs correspondent with the Express since 2005, prior to which he worked at the Press Association as a general news reporter.
William Lewis, editor of the Daily Telegraph, said: "Chris Hope is an excellent journalist who had done a great job covering home affairs for us, as well as playing a key role in a number of our campaigns for our readers. I know he will do very well as Whitehall editor, I am delighted to promote him.
"Tom Whitehead is a talented reporter and I am looking forward to him joining our home affairs team."
In a departure from the Telegraph, deputy editor of the daily title Dan Roberts has left to join Guardian News and Media as head of business. He takes over from Deborah Hargreaves, who has stepped down from the role, which oversees business coverage across The Guardian, The Observer and guardian.co.uk.