The legal dispute involves two divergent views of the role of the executive branch and the court system.
The federal government maintains the president alone has the power to decide who can enter or stay in the United States, while states suing Trump say his executive order is unconstitutional.
Seattle US District Judge James Robart, who on Friday temporarily blocked Trump's order, has said a judge's job is to ensure that an action taken by the government "comports with our country's laws."
The lawyers said today the travel ban was a "lawful exercise" of the president's authority to protect national security and said Robart's order that put the policy on hold should be overruled.
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The filing with the San Francisco-based 9th US Circuit Court of Appeals was the latest salvo in a high-stakes legal fight surrounding Trump's order.
Washington state, Minnesota and other states say the appellate court should allow a temporary restraining order blocking the travel ban to stand as their lawsuit moves through the legal system.
The appeals court over the weekend refused to immediately reinstate the ban, and lawyers for Washington and Minnesota argued anew on Monday that any resumption would "unleash chaos again," separating families and stranding university students. The Justice Department responded that the president has clear authority to "suspend the entry of any class of aliens" to the US in the name of national security.
It said the travel ban, which temporarily suspends the country's refugee program and immigration from seven countries with terrorism concerns, was intended "to permit an orderly review and revision of screening procedures to ensure that adequate standards are in place to protect against terrorist attacks."
The Seattle judge's ruling triggered a Twitter rant by the president.
On Sunday, Trump tweeted, "Just cannot believe a judge would put our country in such peril. If something happens blame him and court system. People pouring in. Bad!" Whatever the appeals court decides, either side could ask the Supreme Court to intervene.
It could prove difficult, though, to find the necessary five votes at the high court to undo a lower court order; the Supreme Court has been at less than full strength since Justice Antonin Scalia's death a year ago.
How and when a case might get to the Supreme Court is unclear. The travel ban itself is to expire in 90 days, meaning it could run its course before a higher court takes up the issue. Or the administration could change it in any number of ways that would keep the issue alive.