But he decided to leave the company to “find his voice”.
Ms Kunsberg is now taking over after a staff reshuffle at the broadcaster.
Welcoming the news, Ms Kunsberg wrote on social media in March: “I am very happy to say that I will be the political chair of the BBC on Sunday mornings from September.
“It’s a real honor and a real pleasure to work with an incredible team on the series, I can’t wait to get started!”
“It’s as simple as that – you knock on the door and people tell you something. A lot of politicians also know it’s coming, but they don’t necessarily want to say it publicly.
“Clearly our decision to leave the EU is the biggest political decision we will make as a country in decades.
“Nobody knows what the consequences will be, good or bad, so I think most people will accept that, which is more important than any choice in the general election.
“Of course, those are huge stories too – you’ve seen politicians trying to fix it, but they don’t really know how to do it.
“We are also in a time when the authorities have been so often wrong that it is normal as a political journalist to say ‘I don’t know’.”
The first “Sunday with Laura Kunsberg” will air on BBC One on September 4 at 9am.