Chris Philp said the sting operation set up Led By Donkeys did not expose any rule breaking but was "pretty unedifying".

Former chancellor Kwasi Kwarteng also fell for the hoax, saying he "wouldn't do anything less than for about 10,000 dollars a month" when asked what he would expect to be paid to advise a non-existent firm in South Korea.

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There is no accusation of wrongdoing, with MPs permitted to seek employment outside of parliament.

However policing minister Mr Philp admitted it was a "pretty unedifying spectacle".

"It is very important that we have transparency over these things. Had that thing we saw been real, then any agreement they would have entered into would have been completely disclosed to the public," he told Sky News.

Asked if Mr Hancock was worth £10,000-a-day, he replied: "I wouldn't pay him £10,000."

Led By Donkeys, an anti-Brexit group, said it created a sham company called Hanseong Consulting, set up a website and paid for a so-called "fake virtual office" in the South Korean capital Seoul.

When asked whether he had a daily rate during an online "interview", Mr Hancock said: "I do, yes. It is 10,000 sterling."

The Conservative former health secretary, who was stripped of the party whip by Prime Minister Rishi Sunak after he was announced as a contestant in last year's series of ITV reality programme I'm A Celebrity... Get Me Out Of Here!, later said he had an hourly rate of "around £1,500".

A spokesman for Mr Hancock said the West Suffolk MP had "acted entirely properly and within the rules".

Former Tory chancellor Mr Kwarteng, when asked the same question, said: "I would say as an MP, obviously I don't need to earn a king's ransom.

"But I wouldn't do anything less than for about 10,000 dollars a month."

Told by a fake employee of the company they were considering offering between £8,000 and £12,000 per day, with the intention for him to attend six board meetings a year, Mr Kwarteng said: "OK yes, we're not a million miles off. We can work with the numbers."

The sting has brought back into the spotlight the issue of MPs' second jobs, which are allowed as long as any extra earnings on top of their £84,144 Commons salary is transparently noted in the register of interests.

However, Labour's Emily Thornberry said MPs are already "really well-paid" and questioned why the likes of the two former ministers "need to go grovelling around the world trying to get more money".

She said while the details of extra earnings must be declared, "the details of these conversations" regarding fees are kept secret.

"They didn't do anything wrong. It's within the rules. But it shouldn't be within the rules," she said.

The Opposition party has pledged to ban second jobs, though Ms Thornberry suggested there should be some exceptions such as public sector and media work - defending David Lammy's outside earnings as "engaging in political debate".

She said Labour will set up an alternative system with an independent commission "that will look at standards" and make sure there are clear rules.

"When this stuff happens, it affects all of us. All MPs are tarred with this and it's not right, it's not fair."