The US supreme court is set to begin action on Donald Trump’s travel ban, leading to a potential final resolution to the defining legal battle of the administration’s early days.

The nation’s highest court is considering an appeal by the Trump administration to allow the president’s controversial order restricting travel to the US from six Muslim-majority countries. Two federal appeals courts have temporarily blocked the revised presidential order by Trump that sought to suspend the issuing of visas from the countries and freeze the US refugee resettlement program.

The appeals court rulings upheld decisions by district judges in Maryland and Hawaii, who ruled against the federal government in March.

The Trump administration appealed these orders to the supreme court, and on Wednesday made its final submissions, calling on the court to temporarily overturn the lower court stays until a full hearing later in the year.

The nine justices are set to meet for their final closed-door conference of the court’s term on Thursday morning, meaning the pathway to a resolution of the case could be announced at any point.

There are a variety of avenues the court could decide to follow that would lead to different outcomes and timeframes for the administration. Firstly, the court needs to decide whether it will hear the appeal at all by granting a certiorari writ. This requires that four justices agree the case should be included on the court’s docket in the first place.

If the court does not grant certiorari, the stays of the lower courts will stand, and Trump’s ban will remain blocked until the district judges proceed with the full cases in the lower courts.

But if the supreme court does decide to take the case on, which seems likely given the high-profile nature of Trump’s executive order, the court would then be in a position to move the case forward.

It could grant the government’s application to overturn the stays or it could keep the stays in place; in both scenarios it would then set a date for a full hearing in court.

If the justices decide on the former, which would require a majority decision, it would effectively allow Trump’s ban to come into effect, and potentially remain in effect throughout the summer.

If it decides on the latter, the court could either decide to hear arguments in October, when it reconvenes after summer recess, or extend the new term and hear arguments in the weeks ahead.

Extending the court’s term, said Carl Tobias, a professor of law at the University of Richmond, would be an extremely rare move, but not one without precedent. In the Pentagon Papers case decided in 1971 the court extended its term to hear arguments in summer.

The court could also simply decide to take the case up in October and not rule on the government’s bid to overturn the stays. Such an outcome would likely be seen as a victory to the plaintiffs in the original cases – a group of Democratic states in the Hawaii case and a collective of migrant rights and legal advocacy groups in the Maryland case – as the stay on Trump’s order would remain throughout the summer, potentially damaging the government’s case further down the track.

Although the supreme court’s term ends next week there may not be any decision announced at all on Thursday and the wait may continue for days.