The shadow Brexit secretary, Keir Starmer, has said Labour will take “every possible step” to compel Theresa May to publish the full legal advice behind her Brexit deal, after accusing the government of ignoring the will of parliament.

“Ministers should be under no doubt that we will take every possible step to force the legal advice to be made available. The government can’t run away from parliament forever,” Starmer said.

Labour has been stepping up its efforts to force the publication of the legal advice after the government made clear it intended to publish only a summary.

On 13 November Labour used an arcane parliamentary procedure known as a humble address to try to force the government to publish the full legal advice from the attorney general, Geoffrey Cox.

MPs refused to accept a concession offered during the debate by David Lidington, who promised the government would publish “detailed legal analysis of the meaning and implications of the agreement” and said Cox would take questions in the Commons.

Starmer insisted that was not sufficient, and Labour went on to win the vote with the support of the Democratic Unionist party (DUP), as Conservative MPs abstained to avoid an embarrassing defeat.

Challenged by Jeremy Corbyn at prime minister’s questions, May confirmed in the House of Commons on Wednesday that she did not intend to publish the full document – which cabinet sources suggest is damning about the Irish backstop.

May said: “We have said and been clear that we will make available to members a full, reasoned position statement laying out the government’s legal position on the withdrawal agreement, and the attorney general is willing to assist parliament by making an oral statement and answering questions from members.

“But as regards publication of the full legal advice, the advice that any client receives from their lawyer is privileged. That is the same for government as it is for any member of the public.”

Labour reacted with fury, and Starmer later asked the Speaker, John Bercow, whether Lidington, who led the debate for the government, could be in contempt of parliament. Bercow suggested Starmer write to him – the next step in the procedure – and added that he “will not linger” in making a judgment.

At best, Labour’s effort to force the publication of the legal advice is likely to disrupt Downing Street’s careful timetable for the days remaining before the 11 December meaningful vote. But punishments for contempt of parliament can include a temporary suspension.