Australia’s ambassador to the United States, Joe Hockey, should be recalled to answer questions about his role in the Bell Group saga, the Greens justice spokesman has said.

Nick McKim made the call on Tuesday, as his party gathered support for a Senate inquiry into the alleged deal between the federal and West Australian governments which would have cost the tax office $300m had it not won a high court case disputing the constitutionality of a WA law to implement the deal.

The inquiry is now likely, as Labor and the Greens have attracted crossbench support from Derryn Hinch and Rodney Culleton for the move.

McKim told ABC radio that rather than a quick-fire inquiry it could take weeks to deal with the issue because the attorney general, George Brandis, had thrown so many of his federal and WA Liberal colleagues “under the bus” in his explanation to the Senate on Monday.

On Monday Brandis said he did not believe there was a deal with the Western Australian government for the commonwealth not to run a particular constitutional argument to defeat the WA law.

Brandis said he had no personal knowledge of discussions between the former federal treasurer Hockey and the WA treasurer, Mike Nahan, in 2015 and that correspondence between them did not prove there was an agreement. McKim said Brandis’s 30-minute statement was “instructive for what it did not contain”.

McKim said it “didn’t address the allegation [Brandis] effectively told [then solicitor general] Justin Gleeson to run dead in the high court” and when that point was put to Brandis in question time, he did not answer.

At a doorstop on Monday, Brandis was asked if he had “effectively told Gleeson to run dead”, and he replied: “That is not the truth and he did not.”

McKim said Brandis should prove this by waiving legal privilege over advice and instructions between Brandis and the solicitor general over the Bell litigation.

“If [Brandis] ordered the solicitor general not to use the strongest argument, that would be an extraordinary abrogation of his responsibilities as attorney general.”

The Greens democracy spokesman said that the fact Nahan had told the Western Australian parliament there was a deal demonstrated the need for an inquiry. McKim said the Senate could call Nahan, WA attorney general Michael Mischin, assistant federal treasurer Kelly O’Dwyer, federal social services minister and former WA treasurer Christian Porter and federal finance minister, Mathias Cormann.

He said there was a “very strong argument” to call Hockey back from Washington to answer questions.

“These are very serious allegations, and he should be called back, and George Brandis threw him under a bus yesterday so he may well want to come back and answer these questions.”

A spokeswoman for Hockey told Guardian Australia he would not be making any comment.

The Senate could not compel Porter and O’Dwyer to appear because they are in the lower house, but McKim said Labor and the Greens’ joint motion would compel Brandis and Cormann to appear.

He said he hoped the motion would pass.

On Tuesday Nick Xenophon told Guardian Australia he opposed the inquiry and a spokesman for Pauline Hanson said One Nation also opposed it.

But a spokeswoman for senator Rodney Culleton said he supported the inquiry, meaning One Nation will split on the question.

Senator Derryn Hinch told Guardian Australia there were still “lots of questions [that] need answering” and he would support the inquiry.

On Monday Labor’s leader in the Senate, Penny Wong, told Lateline that since Brandis had said there was no deal, while Nahan said there was, “both sets of statements … cannot be true”.

Wong argued Brandis and Nahan were also at odds about the role of O’Dwyer and Porter in discussions between the federal and state government.

She argued that Brandis and the prime minister had not answered the question about whether the solicitor general was instructed not to use the particular constitutional argument that won the case for the ATO.

Asked about the fact that Gleeson did eventually run the argument, Wong replied: “The fact that George Brandis was dissuaded from what was a dishonourable path by an honourable man does not absolve Senator Brandis of responsibility.”