With the ban suspended since Friday, the legal battle has moved to San Francisco, where Justice Department lawyers defended the executive order and said the nationwide injunction against it was "vastly overbroad."
A hearing has been set in the case for Tuesday at 3:00 pm (2300 GMT).
Meanwhile two new polls show that a majority of Americans now oppose the ban, which sparked chaos at US airports -- findings that Trump angrily dismissed as media lies.
"Any negative polls are fake news, just like the CNN, ABC, NBC polls in the election," he said on Twitter. "Sorry, people want border security and extreme vetting."
Trump, who paid a visit to US Central Command in Tampa, Florida, earlier in the day once again roundly condemned the media -- accusing them of downplaying the terror threat his administration has cited to justify its travel ban.
"Radical Islamic terrorists are determined to strike our homeland as they did on 9/11, as they did from Boston to Orlando to San Bernardino, and all across Europe," the president told military personnel.
"And in many cases the very, very dishonest press doesn't want to report it," he claimed. "They have their reasons, and you understand that."
Asked to explain Trump's comments, White House spokesman Sean Spicer promised to "provide a list" of attacks that had been "underreported."
"Protests will get blown out of the water, and yet an attack or a foiled attack doesn't necessarily get the same coverage," he said.
Court 'erred'
A federal court that temporarily rolled back Trump's directive "erred in entering an injunction barring enforcement of the order," the government told the Ninth Circuit Court of Appeals in asking that the ban be reinstated.
"But even if some relief were appropriate, the court's sweeping nationwide injunction is vastly overbroad," it said.
The president's executive order summarily denied entry to all refugees for 120 days, and travelers from Iran, Iraq, Libya, Somalia, Sudan, Syria and Yemen for 90 days -- a move critics charge will damage US interests. Refugees from Syria were blocked indefinitely.
But the Justice Department argued that "the executive order is a lawful exercise of the president's authority over the entry of aliens into the United States and the admission of refugees."
The Justice Department defended the ban as a means to undergo "an orderly review and revision of screening procedures to ensure that adequate standards are in place to protect against terrorist attacks."
It stressed that non-US citizens seeking to enter the United States for the first time have no constitutional rights.
Mounting opposition
On Friday in Seattle, a federal judge ordered the temporary nationwide suspension of the president's order, allowing the many travelers who were suddenly barred from US soil to start trickling back in.
On Sunday, the appellate court refused to overrule the federal judge.
Attorneys general for the states of Washington and Minnesota, which won the temporary stay of the ban, have asked the appeals court to refuse to reinstate it. They were backed Monday in a court brief filed by 16 other US states.
The case may ultimately reach the US Supreme Court.
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Several legal and rights groups have filed in support of the states, including the Southern Poverty Law Center, Americans United for Separation of Church and State and the American Civil Liberties Union.
And in an additional blow, a slew of Silicon Valley giants led by Apple, Facebook, Google, Microsoft and Twitter have filed a legal brief in support of the lawsuit.
The 97 companies speaking out against Trump's travel ban said it harms recruiting and retention of talent, threatens business operations and hampers their ability to attract investment to the United States.
A group of prominent Democrats including former secretaries of state John Kerry and Madeleine Albright added their voices to the criticism.
"Reinstating the executive order would wreak havoc on innocent lives and deeply held American values," they said in a filing.
Specifically, the Democrats said Trump's travel ban could endanger US troops in the field, disrupt counterterrorism cooperation and feed Islamic State group propaganda.
Arrivals
Spicer insisted the administration was "absolutely not" thinking about withdrawing its order.
On Trump's plan to introduce so-called "extreme vetting" at US borders, he said: "Once we win the case, it will go right back into action."
But top Republicans have also shown renewed signs of discomfort with the president as the controversy escalates.
Trump had blasted James Robart, the Seattle federal judge, in a series of angry tweets.
"Just cannot believe a judge would put our country in such peril. If something happens blame him and court system. People pouring in. Bad!"
With the ban suspended, travelers from the targeted countries holding valid visas have begun arriving on American soil.
In New York, 33-year-old Sudanese doctor Kamal Fadlalla rejoiced -- after a week blocked in his home country, he was back in the Big Apple with friends and colleagues.
"It feels great," Fadlalla told AFP on Sunday at John F. Kennedy International Airport. "It was a tough week actually."
The State Department has said visa holders from the seven countries are allowed to travel to the US as long as their documents have not been "physically canceled."
