Clamour is growing for David Cameron to face scrutiny from MPs about his lobbying of Government, as Labour launches a bid to force a Commons inquiry. Sir Keir Starmer’s party will use its Opposition Day Debate on Wednesday to call a binding vote on a motion to set up a “full, transparent” parliamentary probe into the Greensill controversy. Labour proposes creating a new cross-party select committee to conduct the inquiry, which would have the power to call witnesses to give evidence and face questions. A party source said it was imperative that Mr Cameron was quizzed in public and not only behind closed doors. The former prime minister is currently expected to give evidence privately to an independent probe looking at his lobbying activities, launched by the Government on Monday, after his spokesman confirmed he would “be glad to take part” in it. Downing Street announced that lawyer Nigel Boardman, a non-executive board member of the Business, Energy & Industrial Strategy department, would review how collapsed lender Greensill secured Government contracts and the role of lobbying by the finance firm’s representatives, including Mr Cameron, aimed at Government. Mr Cameron lobbied Rishi Sunak and two other Treasury ministers last year on behalf of Greensill, it has emerged, while he also organised a “private drink” between Matt Hancock and Mr Greensill. Rachel Reeves, Labour’s shadow chancellor of the Duchy of Lancaster, poured scorn on the lawyer-led inquiry, which she claimed “has all the hallmarks of a Conservative cover up” and amounts to the Conservatives “marking their own homework”. The public “deserve answers” about the deep access Greensill was reportedly granted to Whitehall departments under Mr Cameron’s administration and the contact Mr Cameron has had with ministers since he became an adviser to the firm. She said: “Any Conservative who wants to stop the cronyism rampant in their party and in Government must vote with Labour this week to uncover once and for all the truth behind this scandal.” The Opposition party’s demand that the Chancellor, the Health Secretary, the Prime Minister and special advisers also face scrutiny as part of the proposed probe is likely to dampen Tory support for the idea, however. Alternative options for a parliamentary inquiry into the lobbying row engulfing Mr Cameron and the collapsed lender Greensill are being discussed by MPs. Chris Bryant, Labour chairman of both the standards and privileges committees, has called for a joint inquiry to be hosted by a group of relevant select committees. He said that the benefit of an MP-led probe included powers to subpoena witnesses and evidence, and offer the cloak of parliamentary privilege to whistleblowers. “Whatever you say in committee you can't be sued for. This applies to courts, tribunals and parliamentary proceedings. It’s a really important aspect of the way we do our business. It helps you get to the truth.” Senior Conservative MP Sir Bernard Jenkin, chairman of the liaison committee, expressed his openness to the idea of a parliamentary committee. He told The Telegraph: “There needs to be some form of inquiry which will learn some lessons from the Greensill controversy and the many other concerns that have been raised over decades about inappropriate relationships between Government and business. This didn’t start under the Conservatives.” Changes to the civil service code and ministerial code are needed, he said, adding: “What matters most is not who is doing the lobbying but how the system responds to inappropriate lobbying.” Mr Cameron broke his silence on Sunday and stressed that his request of Mr Sunak, to boost Greensill’s access to a Government coronavirus funding scheme, was rejected. “The outcome of the discussions I encouraged about how Greensill’s proposals might be included in the Government's CCFF [Covid Corporate Financing Facility] initiative-and help in the wake of the coronavirus crisis-was that they were not taken up,” he said. The former prime minister said there were “important lessons to be learnt”, and that he should have acted differently “so there can be no room for misinterpretation”. In a statement he said by making representations to Government on behalf of Greensill, he “was breaking no codes of conduct and no government rules”. He cautioned that “the outcome of the discussions I encouraged about how Greensill’s proposals might be included in the Government's CCFF [Covid Corporate Financing Facility] initiative-and help in the wake of the coronavirus crisis-was that they were not taken up”. Mr Cameron also sought to clarify the use of text message and email, as a means of communication. “I understand that concern, but context is important: at that time the Government was-quite rightly-making rapid decisions about the best way to support the real economy and welcomed real time information and dialogue,” he said.