Allen, who has been with BBC Radio 5 Live since the station started in 1994, said he thinks the votes for Brexit and Donald Trump have made him nervous.
He told the Radio Times: “There are troubles, obviously there are troubles. An unholy thought struck me on the way in here - we had a vote for Brexit and we went for Brexit, we had a vote for Trump and we went for Trump. If we had a vote for the BBC, which way would it go? Let’s make sure it goes our way.”
Allen has reunited with Jane Garvey, the first ever voice on 5 Live, for a new show on the station as it shakes up its autumn schedule.
The award-winning double act, who fronted 5 Live Breakfast and 5 Live Drive for 13 years, are hosting a new Sunday evening show about people in the news.
Garvey added: “What organisation is healthy and happy? People have been carping about BBC management for as long as I have worked for the BBC, which is getting on for 25 years. That is not going to change. But am I glad I’m here? Yes.”
Allen added he is thrilled by the success of Emma Barnett since she began presenting a new morning show on 5 Live following his departure.
He said: “I’m delighted for Emma. She’s a formidable lady. It was time for me to leave. I’m sure the boss thought the same. It’s a lovely job but people do tend to cling on to it.”
Garvey called for more female presenters on the radio, saying: “Emma’s a force of nature. I’m full of admiration. I think we need more women on the radio, there’s no doubt about it. So I’d be rather hypocritical if I didn’t think it was great.
“She’s energetic, she’s got fire in her belly. The worst reaction for any presenter is no reaction at all. Indifference is a nightmare.”
Garvey, who also presents Woman’s Hour on BBC Radio 4, admitted some listeners resisted her when she joined the show. “Some listeners had concerns at first. I didn’t sound like other presenters on Radio 4,” she said.
“I still don’t sound like a lot like them - and I don’t want to sound like some of the other presenters. But the Woman’s Hour audience has gone up. Some people hate me and love Jenni (Murray); and some people are the other way round.”
:: The full interview is in the Radio Times, on sale now.