Shadow cabinet office minister Rachel Reeves also warned that the lobbying scandal threatening to engulf Boris Johnson’s government was leading to the “erosion of trust in our politics”.
A leading Tory MP has also warned the prime minister he risks losing the electoral gains he made against Labour if he does not clean up the “shameful” Greensill affair.
The row erupted after it emerged that Mr Cameron had texted the chancellor Rishi Sunak on behalf of Greensill Capital.
In the latest twist in the story, which has sparked a series of inquiries, The Sunday Times reports that Mr Cameron also lobbied the NHS on behalf of the company just weeks into the coronavirus pandemic.
Ms Reeves told the BBC’s Andrew Marr Show: “Tory sleaze is back, it is bigger than ever, and we are seeing the erosion of trust in our politics because of the behaviour by a few at the top of the Conservative Party.
“I don’t want any stone to go unturned in these investigations, but let’s be clear – standards have fallen so far in the last 11 years under this Conservative government.”
She added: “What we want to see – we wanted this week, and that is why we had a vote on it – to have a proper inquiry.
“Not chaired by one of Boris Johnson’s friends, a very close friend of the Conservative Party is Nigel Boardman, who is doing the review, but a proper independent inquiry that has teeth and has the chance to make recommendations on how to clean this up.
“Because this isn’t just about Greensill, Greensill is the tip of the iceberg.”
However. she dismissed suggestions that the situation could be compared to that of the former Labour Welsh first minister Carwyn Jones, who took a private sector job despite concerns from officials.
“With lobbying it takes two to tango,” she said. “For every person doing the lobbying there is someone in power today who is willing to take those calls, to have those drinks, to push those boundaries.
“And that is what is happening in this government.”
On the same programme the environment secretary George Eustice described Mr Cameron’s actions as “acceptable”, saying he had acted within the rules.
“He has not broken any of the rules. It is acceptable,” he said.
He added: “I think always on these things the issue is less about who someone might have spoken to, the issue always is how did a minister act after that conversation?
“And I think it is very clear to me, from what I can see that the Chancellor has not acted at all improperly.
“He flagged it, he looked at it and nothing was done on that case.”
Asked about Mr Cameron’s bid to influence ministers, the Environment Secretary added: “He didn’t get anything for it and the company went bust.”
Mr Eustice added that he had last spoken to Mr Cameron probably a “few months ago” and that they had not discussed Greensill.
But he said that “tweaks” to the system by which ministers declare outside interests could be made in the wake of the scandal, in an interview with Times Radio.
He said: “There’s quite a lot of declaration of interest, or potential interest, that goes on at the moment. It’s quite a thorough system that we’ve got and it generally works quite well. But that’s not to say that if following this episode there are certain lessons to be learned or tweaks or changes that could be made that we shouldn’t look at this.”