A federal appeals court in San Francisco is hearing arguments for and against President Trump's travel ban.
The executive order temporarily banned entry for all refugees and visitors from seven mainly Muslim countries.
Lawyers for the US Justice Department were first to make their case, urging the appeal judges to reinstate the order because Congress has authorised the president to control borders.
The orders were halted a week ago after two states challenged it.
They had said it was unconstitutional and harmed the interests of the states.
Whatever the 9th US Circuit Court of Appeals decides - and a ruling is unlikely to come on Tuesday - the case will probably end up in the Supreme Court.
Media captionWhy did an Iraq veteran give a stranger his Purple Heart medal?
Media captionLancaster, Pennsylvania: America's refugee capital
A federal judge temporarily blocked Donald Trump's order last week.
As a result, people from seven countries - Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen - with valid visas were able to travel to the US again.
Washington state, Minnesota and other states want the appeals court to permit the temporary restraining order to stand as their lawsuit works its way through the courts.
A 15-page brief issued by the Justice Department on Monday night argued the executive order was a "lawful exercise of the president's authority" and not a ban on Muslims.
In Friday's case, the Justice Department had argued that states did not have the authority to challenge a presidential executive order.
Lawyers for the states of Washington and Minnesota had argued that the ban was unconstitutional because it denied people with valid entry documents the right to travel without due process.
It also violated freedom of religion rights by appearing to target Muslims, they said.
What happens now - BBC's Anthony Zurcher, Washington
The next step is for briefs to be filed by both sides for a formal review of Judge Robart's suspension on Monday. The Justice Department could have appealed directly to the Supreme Court on an emergency basis, but it chose not to since the appeal court is moving fairly quickly.
If the appeal court decides the stay is valid - perhaps as early as next week - then a Supreme Court appeal is almost certain.
In the meantime, everything is on hold. US immigration processes continue as they did before Mr Trump issued his executive order.
If it looks like this is bogging down, the president might eventually decide to modify the order rather than try to defend its legality. That's probably the most prudent course, but he's a stubborn man.
Iraq, one of the countries named in the ban, has praised the revocation of the travel ban as a "move in the right direction", Reuters reported.
Iran has also responded to Judge Robart's ruling by saying it would allow a US wrestling team to compete in a World Cup event it is hosting later this month.
Image copyrightAPImage captionUS wrestlers will now compete in Iran, which has relaxed its own retaliatory ban
The US wrestlers were initially denied visas after Iran said it would ban Americans in retaliation for Mr Trump's order.
However Mr Trump has called Judge Robart's ruling "ridiculous", described him as a "so-called judge" and vowed to restore the ban.
Republican Senate leader Mitch McConnell told CNN it was "best to avoid criticising judges individually".
Judge Robart has served on the federal bench since 2004 after nomination by President George W Bush.
Media caption"We are very excited, we are very happy" - Iraqi national flies to the US with his family
Friday's ruling has also seen visa holders from the affected nations scramble to get flights to the US, fearing they have a slim window to enter.
The State Department has been reversing visa cancellations and US homeland security employees have been told by their department to comply with the ruling.
Customs officials told airlines that they could resume boarding banned travellers. Qatar Airways, Air France, Etihad Airways, Lufthansa and others said they would do so.
The ban caused confusion at US and foreign airports when it came into force.
Media caption"I have five things to say to you": Immigrant rights campaigner Thanushka Yakupitiyage
It envisages a 90-day visa suspension for anyone arriving from Iraq, Syria, Iran, Libya, Somalia, Sudan and Yemen.
The directive also suspends the US Refugee Admissions Programme for 120 days, and places an indefinite ban on Syrian refugees.
It has led to protests in US cities and around the world.
