Mr O’Brien told listeners: “I don’t know whether it would work legally, but morally and indeed in terms of messaging, it struck me as a brilliant idea to fine them both
“It’s not a life-changing fine, it’s £100 I think.
“But it would send a message to everybody that nobody is exempt.
“I know what you’re thinking: Barnard Castle, but that ship has sailed.”
READ MORE: Corbyn explodes as he sulks about media ‘obsession’ with Russian leak
He continued: “That appalling insult to every single human being in the United Kingdom, let alone the ones who have suffered bereavement and loss during this coronavirus crisis.
“But Dominic Cummings’ contempt for you and yours is now just a matter of established historical fact.
“We can’t change it. That is the contempt in which he holds you and me, and your mum and my mum.
“Park that, but maybe don’t move on. There’s nothing you can do about it.”
Currently first time offenders will be fined £200 to failing to comply with the guidance.
This can be as much as £10,000 for serious breaches of regulation.
Labour leader Sir Keir Starmer did not call for his predecessor to be fined, but insisted that Mr Corbyn should have obeyed the rules.
He said: “It’s not for me to decide who should be fined, but it is for me to say that everybody should follow the rules.”
Downing Street has said it would be up to the police to decide whether any action should be taken against Mr Corbyn.