The government is facing a clash with some of the country's most senior judges who will this week attempt to force ministers to relinquish control of the running of the supreme court.

The constitutional dispute between the judiciary and the executive focuses on who should have the power to appoint the chief executive of the UK's highest court – ministers or the judges who sit on it.

In his first political intervention since stepping down as the president of the supreme court, Lord Phillips of Worth Matravers will try to change the law to force Chris Grayling, the justice secretary and lord chancellor, to renounce his power of patronage over the appointment.

The challenge will come in an amendment that Phillips has tabled to the crime and courts bill, which calls for the power to appoint the chief executive to be transferred to the president of the supreme court.

Lord Neuberger, who currently holds the post, has told ministers he supports such a transfer.

The amendment has the backing of the House of Lords constitution committee and is expected to be debated on Tuesdayand taken to a vote.

In a letter sent last week, Lady Jay, the chair of the committee, told justice minister Lord McNally: "The [supreme] court's independence, and the perception of its independence, requires that the chief executive owes her primary loyalty to the president of the court, rather than a minister."

"The chief executive is best placed to determine the staffing requirements of the court, and ... should as a matter of constitutional principle be accountable to the president and to parliament for such issues, and not a minister."

The judges are determined there should be a complete separation of powers between the judiciary and the executive arm of government. The crossbench peer Lord Pannick, who tabled the amendment with Lord Philips, said: "For the president of the supreme court to have the responsibility for appointing the chief executive would emphasise to all concerned that this is an independent institution.

"The existing appointment provision [has] led more than once to confusion in parts of the government machine that the chief executive should in some sense be acting at the behest of ministers."

Tuesday is also the day the government's Commission on the Bill of Rights will produce its long-awaited report on whether or not the Human Rights Act should be replaced by new legislation.

The document will be handed over to Grayling and the deputy prime minister, Nick Clegg, shortly before it is published.

There is widespread expectation that the commission's members, who include the Liberal Democrat Lord Lester, Labour's Lady Kennedy and the Conservative Anthony Speight QC, will fail to reach a consensus and end up in political stalemate, unable to agree on a common set of recommendations.

Most human rights groups and representatives of the devolved governments in Wales, Scotland and Northern Ireland are determined to safeguard the existing act. They fear any attempt to replace it with a UK bill of rights would be used as political cover for removing the country from the European convention on human rights.

In the wake of a number of controversial rulings by the European court of human rights in Strasbourg, anti-Europe sentiment is running strongly in the Conservative party.

The supreme court's current chief executive, Jenny Rowe, has been widely praised for helping establish the court after the judges moved out of the House of Lords in 2009 to the former Middlesex Guildhall in Westminster, directly opposite parliament.

When the issue of the supreme court's chief executive was raised in the Lords during a debate earlier this month, the Conservative minister, Lord Ahmad of Wimbledon, resisted suggestions that the running of the court should be handed over entirely to the judiciary.

He said the government "fully and utterly respect the independence of the judiciary" but added it "retained a fundamental concern with regard to ... proper lines of accountability which need to be established so that the elected government are responsible for the proper fiscal and managerial operation of the court".