MPs will examine the gulf between what Tony Blair said in public and in private about invading Iraq. Photograph: Stefan Rousseau/PA

A cross-party group of MPs will make a fresh effort to hold Tony Blair to account for allegedly misleading parliament and the public over the Iraq war.

The move, which could see Blair stripped of membership of the Privy Council, comes as the former prime minister tries to re-enter the political fray, promising to champion the “politically homeless” who are alienated from Jeremy Corbyn’s Labour and the Brexit-promoting government of Theresa May.

The group, which includes MPs from six parties, will put down a Commons motion on Monday calling for a parliamentary committee to investigate the difference between what Blair said publicly to the Chilcot inquiry into the war and privately, including assurances to then US president George W Bush.

Backing the motion are Alex Salmond, the SNP MP and former first minister of Scotland; Hywel Williams, Westminster leader of Plaid Cymru; and Green party co-leader Caroline Lucas.

Senior Tory and Labour MPs are also backing the move, which reflects widespread frustration that the publication of the Chilcot report in July, after a seven-year inquiry, did not result in any government action or accountability for Blair.

Salmond said some MPs believe that senior civil servants were “preoccupied with preventing previous and future prime ministers being held accountable”. He said: “An example should be set, not just of improving government but holding people to account.”

He pointed to last week’s Observer story revealing that, according to documents released under the Freedom of Information Act, the inquiry was designed by senior civil servants to “avoid blame” and reduce the risk that individuals and the government could face legal proceedings.

Salmond also noted that documents show many officials involved in planning the inquiry, including current cabinet secretary Sir Jeremy Heywood, were involved in the events that led to war.

The new motion will be debated on Wednesday in Commons time allotted to the SNP. It calls on MPs to recognise that the inquiry “provided substantial evidence of misleading information presented by the then prime minister and others on the development of the then government’s policy towards the invasion of Iraq as shown most clearly in the contrast between private correspondence to the United States government and public statements to parliament and people”.

The motion also asks the Commons public administration and constitutional affairs committee to add to its current inquiry into the lessons to be learned from Chilcot “a further specific examination of this contrast in public and private policy and to report on what further action is necessary to help prevent repetition of this disastrous series of events”.

Salmond said the committee could “recommend whatever action it pleases”, including that Blair be stripped of membership of the Privy Council, which formally advises the sovereign and exercises government and judicial functions.

This would be an unprecedented move in respect of a former prime minister, but Williams said: “If he continues to be a member of the Privy Council while there is all this damning evidence against him, what does that say about the institution?”

Williams told the Observer the process used was “scrupulously fair” to Blair because he will have a chance to appear at the committee to defend himself.

Lucas said: “The Chilcot report confirmed Tony Blair lied to the public, parliament and his own cabinet in order to drag us into the Iraq war. Privately, he said he would support Bush ‘whatever’ eight months before the war – everyone else was told war could be avoided.

“Thousands of lives were lost because he put that promise before all the evidence. Yet – despite the damning evidence against him contained in the inquiry’s report – no action has been taken against the former prime minister.”

A spokesman for Blair declined to comment. But, privately, his supporters say similar motions have been tabled before without gaining significant traction among MPs. They said that Chilcot rejected allegations that Blair said one thing in public and another in private.

Appearing before the Commons liaison committee, Chilcot had said: “I absolve him [Blair] from a personal and demonstrable decision to deceive parliament or the public – to state falsehoods, knowing them to be false.”