The government confirmed last week that it would be up to the prime minister to judge whether Rishi Sunak and Matt Hancock had broken the ministerial code during the affair – prompting calls to overhaul the system and make it more independent.
The chancellor and health secretary are facing questions about how they dealt with approaches by former PM David Cameron, who lobbied them to give now collapsed bank Greensill Capital access to taxpayer support.
But writing to MPs, Cabinet Office minister Chloe Smith said that enforcement of the ministerial code was ultimately "the responsibility of the prime minister of the day".
Critics say they are worried that Mr Johnson will simply give his colleagues a clean bill of health whatever an independent inquiry set up by the government finds.
The prime minister pulled a similar manoeuvre last year when an investigation found evidence that the Home Secretary Priti Patel had broken the ministerial code by bullying her colleagues – first sitting on the report for a year and then insisting no action was required.
Now the Liberal Democrats have written to Mr Johnson and urged him to delegate the decisions on whether to launch investigations into his colleagues to an independent figure.
And Labour MP Dan Carden has said the government should go even further on the issue and put the ministerial code into law so that courts could rule in cases like Greensill.
He accused the government "acting with impunity, with genuine accountability nowhere in sight".
"If Boris Johnson is responsible for upholding standards of behaviour, then we’ve got problems," Mr Carden told The Independent.
"It’s no secret that the current rules in place to safeguard integrity in public office aren’t worth the paper they are written on. We need a complete reset to clean up politics and root out conflicts of interest.
"Among other measures, it’s time for the Ministerial Code to become law. The Conservatives simply cannot be trusted to mark their own homework."
Noting that last year alone Transparency International UK counted at least 30 different potential breaches of parliamentary and ministerial rules without any action being taken, Mr Carden added:
"From the Greensill scandal to the multi-million pound contracts handed to Tory friends and donors, it’s no wonder that trust in our political system has collapsed."
In a letter to Mr Johnson seen by The Independent, Liberal Democrat leader Ed Davey and Lib Dem cabinet office spokesperson Alistair Carmichael said the Greensill scandal was "harmful not only because of the wrongdoing which has been exposed in this particular case, but because it undermines the trust of the public in all politicians".
"Our voters have the right to expect that as their elected representatives, we will uphold the highest standards. It is now up to you to take decisive and urgent action to rebuild their trust," they argue.
"In particular we believe the recommendations from the Committee on Standards in Public Life that the Independent Adviser should have authority to decide whether or not to investigate an alleged breach of the Ministerial Code and that they should be able to publish their conclusions, will be a valuable first step towards rebuilding public confidence in the independence of these processes.
"We hope you will see fit to make a statement to the House on these recommendations and other steps you will be taking to restore trust in the conduct of senior members of your Government in the wake of these reports, at the earliest possible opportunity."
In a written parliamentary answer minister, Chloe Smith said the government has "no plans for legislation" to make the code legally enforceable – leaving Mr Johnson as the unlikely last line of defence against perceptions of cronyism.
Both Mr Sunak and Mr Sunak asked civil servants to look into helping Greensill after being approached by Mr Cameron, but deny they have done anything wrong. Campaigners say the rules around lobbying need to be toughened up and enforced independently.
Asked whether powers to enforce the ministerial code could be written into law to be overseen by the courts, the minister Ms Smith said: "The Government has no plans for legislation on this matter.
"The ministerial code is the responsibility of the prime minister of the day and customarily updated and issued upon their assuming or returning to office."
She described the code as "guidance" on "how [ministers] should act and arrange their affairs", adding that it "lists the principles which may apply in particular situations".
The minister added: "It is the prime minister’s responsibility to set standards of behaviour for members of the executive, and to account for the actions of the government."
Transparency International UK chief executive Daniel Bruce told The Independent: “The code of ministerial conduct needs to be put onto statutory footing and the replacement for the PM’s independent adviser and their team needs to have the autonomy and independence to launch and conclude their own investigations into suspected breaches of the ministerial code.
“And they need to have a ladder of sanctions available to them to enable them to take a pragmatic and proportionate approach to any breaches.
“We need to create a separation of the executive from oversight of ministerial behaviour. Recent history has told us time and again that the advisory, voluntary, discretionary approach is under considerable strain. For the electorate to have confidence that those representing them are subject to suitable checks and balances, we need to separate it from the executive and make it fully independent.”
A spokesperson for the Cabinet Office said: "The Boardman Review will be a full and thorough examination of the facts.
"He will have access to all the necessary information required and engage with those involved at the time when decisions were made. The findings will also be made public and presented to Parliament."
Asked about the enforceability of the review's findings on whether the ministerial code was broken, the Cabinet Office spokesperson said they had nothing further to add to the comments by Ms Smith.