Both MPs deny any wrongdoing, as Vicki Young reports
Two former foreign secretaries have been suspended from their parliamentary parties after being secretly filmed apparently offering their services to a private company for cash.
Conservative Sir Malcolm Rifkind and Labour's Jack Straw both say they have broken no rules.
The MPs have referred themselves to Parliament's standards watchdog.
Labour leader Ed Miliband has written to the prime minister calling for a ban on MPs having second jobs.
It is claimed that Mr Straw was recorded describing how he operated "under the radar" and had used his influence to change EU rules on behalf of a firm which paid him £60,000 a year.
On the subject of payment, Mr Straw is heard saying: "So normally, if I'm doing a speech or something, it's £5,000 a day, that's what I charge."
Sir Malcolm is reported to have claimed he could arrange "useful access" to every British ambassador in the world.
The MP for Kensington and chairman of Parliament's Intelligence and Security Committee was recorded saying: "I am self-employed - so nobody pays me a salary. I have to earn my income."
He said his usual fee for half a day's work was "somewhere in the region of £5,000 to £8,000".
MPs' second jobs
Both men defended themselves on appearances on BBC Radio 4's Today programme on Monday morning.
Sir Malcolm said he had "nothing to be embarrassed about". He said the allegations were "unfounded" and he vowed to fight them "with all my strength".
He said he had never accepted an offer from the fake firm, saying it was a "preliminary" discussion "about what they had mind".
Sir Malcolm is paid £67,000 a year and he said telling the company he was not paid a salary was a "silly thing to say".
"Of course I receive a salary as a Member of Parliament but I was referring to my business interests, from none of which I receive a salary. I receive payment for services I provide," he said.
He told the Daily Politics an MP's salary "sounds a lot of money to anyone earning less than that".
But he added: "The reality is that to anyone from a professional or business background earns considerably more than that."
Limiting MPs to their parliamentary salary would be "excluding very large numbers of very able people" who could not "accept such a substantial reduction in their standard of living," he added.
About 200 MPs have business interests, he said, and everything he earns is detailed in the Register of Members' Interests.
Sir Malcolm said he would not stand down as security committee chairman, unless his committee colleagues wanted him to.
"One's got nothing to do with the other," he said. "None of the matters are remotely to do with intelligence or security."
He said he had a letter from Channel 4, accepting he had not offered access to any privileged or secret information.
Mr Straw has suspended himself from the Parliamentary Labour Party, and the party said it was aware of the "disturbing allegations" against him.
The Blackburn MP, who had already announced his intention to stand down in May, said he was "mortified" that he had fallen into the reporters' "trap" but that he had said nothing "improper".
He told Today the language he used had been "not necessarily wrong but could be taken out of context".
During his 36 years as an MP he had been "absolutely scrupulous" about observing the rules, he said, adding that the entire discussion had been around what he would do after leaving Parliament.
He acknowledged he should have postponed the conversation until after 7 May.
Earlier, Mr Straw said he had taken on one consultancy role since his ministerial career ended in 2010, with commodity suppliers, ED&F Man (Holdings) Ltd, saying it was done in accordance with the MPs' Code of Conduct.
Sir Malcolm and Mr Straw both said they had requested copies of the recording transcripts, but that Channel 4 and the Telegraph had not provided them.
Former Prime Minister Tony Blair offered Mr Straw his support, saying he was a "byword for being a hard-working constituency MP and parliamentarian".
Green Party leader Natalie Bennett said her party would "end second jobs for MPs".
Mr Miliband said the allegations against Mr Straw were "disturbing" and called for the issue of second jobs to be settled "once and for all".
Prime Minister David Cameron said he did not favour a complete ban on MPs having other employment, saying Parliament was "enriched" in some cases by members' outside experience.
Describing the reports as "very serious matters", he promised an "immediate disciplinary inquiry" into Sir Malcolm's case.
The Independent Parliamentary Standards Authority, which manages MPs' pay, has recommended a 9% salary rise, but party leaders have said that would be unacceptable.
Phillip Blond, director of the centre-right thinktank ResPublica, said MPs should be banned from having outside jobs - but should also have their pay increased.
"We need to recognise that MPs are at the top of the public service tree, and pay them at the same level as top GPs, top civil servants, top head teachers," he said.
The undercover reporters had created a fictitious communications agency called PMR, which they said was based in Hong Kong.
A statement on Channel 4's website said 12 MPs with "significant outside interests" were invited to apply for jobs with PMR, which had "plenty of money" and wanted to hire "influential British politicians to join its advisory board".
"Not all politicians are for hire," the statement added.
"Half of those approached didn't respond. One said he wanted to check us out in Hong Kong so we took it no further. And another said he just wasn't that interested. Of the others, two stood out - Sir Malcolm Rifkind and Jack Straw."
The documentary, called Politicians for Hire, will be broadcast on Channel 4 at 20:00 GMT.