MPs are to vote on whether to establish a parliamentary inquiry into David Cameron’s lobbying activities for collapsed financial firm Greensill Capital.

A plan put forward by Labour would create a new Commons select committee to investigate lobbying, including the former prime minister’s activities, which would be able to summon witnesses to answer questions in public.

If MPs approve the motion on Wednesday, the cross-party committee would investigate whether current laws are sufficient to prevent “inappropriate lobbying” of ministers and officials.

Labour’s calls for an inquiry intensified on Tuesday after it emerged that the former head of Whitehall procurement became an adviser to Greensill Capital while still working as a civil servant, in a move approved by the Cabinet Office.

Bill Crothers began working for the firm as a part-time adviser to the board in September 2015 and did not leave his role as Government chief commercial officer until November that year.

Shadow chancellor of the Duchy of Lancaster Rachel Reeves urged Tory MPs to back the motion on Wednesday if they want to “stop the cronyism rampant in their party and in government”.

Opposition day motions are not usually binding on the Government, but because this is calling for Parliament rather than ministers to establish an inquiry, it would lead to the creation of the committee.

The proposed committee would have the power to “send for persons, papers and records” – giving it the ability to summon Mr Cameron and ministers such as Chancellor Rishi Sunak to give evidence and answer questions from MPs.

Mr Cameron sent text messages to the Chancellor and reportedly took Health Secretary Matt Hancock for a “private drink” with his employer Lex Greensill as part of his lobbying activities.

The Conservatives cannot be trusted to mark their own homework on this

Labour has claimed that a separate independent review established by Boris Johnson, which will be led by lawyer Nigel Boardman, will be a “Conservative cover up”.

Prime Minister Boris Johnson said that the lawyer will be given the “maximum possible access” to get to the bottom of what happened, adding that he would like the review to be “done quickly”.

A Cabinet Office spokesman said the “wide-ranging” independent review into the financial firm, announced on Monday, will also consider the situation of Mr Crothers, who later became a director at Greensill.

Mr Crothers was still working as a civil servant when he took up the advisory role at Greensill, a move which he said was approved by a conflicts of interest policy and “supported by the Cabinet Office leadership”.

It comes as a photo of Mr Cameron and Mr Greensill during a business trip to Saudi Arabia in January last year was released, a visit which saw the pair meet with crown prince Mohammed bin Salman.

Ms Reeves 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 Conservatives cannot be trusted to mark their own homework on this.

“The Boardman investigation has all the hallmarks of a Conservative cover up – the British public, especially those with their jobs on the line as a result of Greensill’s collapse, deserve answers.”

Former attorney general Dominic Grieve QC, who was Mr Cameron’s principal legal adviser between 2010 and 2014, said the public were “increasingly cynical” about the behaviour of politicians.

“If you have been a minister of the crown, and you have held high office, then I think to turn oneself into a paid lobbyist is an unfortunate thing to do,” the former Conservative MP told Channel 4 News.

The motion will not pass without the support of Conservative MPs, something that is unlikely on an opposition proposal, but the move will maintain pressure on ministers over the affair.

The former prime minister, who sent text messages to the Chancellor and contacted other ministers, has admitted that he should have communicated with the Government “through only the most formal of channels”.

Greensill Capital, which collapsed into administration in March, was the biggest backer for Liberty Steel, a firm which employs thousands of workers around the UK and now faces an uncertain future.