“This is not a secret court, an inquisition from medieval times. This is the United States of America,” U.S. District Judge Alvin Hellerstein said at a hearing in New York. “You gotta tell a person what he’s done.”

At issue in the New York and Colorado cases is the administration’s use of the Alien Enemies Act of 1798 to deport alleged members of the Venezuelan gang Tren de Aragua, bypassing normal immigration procedures. Trump in mid-March signed a proclamation declaring the gang equivalent to a hostile foreign government in wartime and ordering government agencies to immediately remove its alleged members. Hours after the proclamation was made public, immigration officials launched planes moving more than 200 migrants, most of them Venezuelans, to a notorious prison in El Salvador.

More people currently detained in U.S. facilities have gone to court in a bid to block the administration from deporting them in the same fashion.

In one of those cases, U.S. District Judge Charlotte N. Sweeney in Colorado expressed serious doubt Tuesday about the government’s claims that current events constitute “irregular warfare” in which gang members are immigrating to the U.S. to commit murders, kidnappings, drug trafficking and other offenses.

Sweeney all but said Trump’s invocation of wartime authority for the gang deportations was unlawful. To use the Alien Enemies Act, Trump would have to establish the U.S. is in a state of war, the judge said.

To the extent Trump’s proclamation “relies on the Act’s invasion and incursion provisions to justify its removal powers, it does so improperly,” the judge wrote.

Sweeney blocked the government from using the act to remove anyone detained in the state without at least 21 days notice.

The Supreme Court earlier this month said that detainees are entitled to notice of pending removal from the country and an opportunity to challenge their deportations but didn’t specify what notice was required, leaving that to the lower courts to decide.

In New York, Hellerstein said he was skeptical the government could remove Venezuelan migrants just based on their alleged membership in the gang Tren de Aragua. Hellerstein ripped into the government’s current method of notifying detained migrants that were slated for deportation, saying the notices needed to come sooner, be in their native language and include more details about how to get a lawyer.

The judge also raised concerns about the treatment of detainees who had been shackled and deported to a notorious maximum-security prison in El Salvador in March, saying that future detainees should be afforded more protections. The U.S. has paid El Salvador’s government at least $6 million to incarcerate deported alleged gang members.

Nothing in the president’s proclamation allows the U.S. “to hire a jail in a foreign country where people could be subjected to cruel and unusual punishment not allowable in a U.S. jail,” Hellerstein said.

At the end of the hearing, the judge extended a temporary restraining order barring the removal of detainees being held in his jurisdiction without due process while he decides on a longer-term injunction.

In the case involving Abrego Garcia, who was mistakenly deported to El Salvador along with hundreds of other migrants in March, U.S. District Judge Paula Xinis expressed exasperation with the administration’s repeated delays in providing details about its efforts to return him. Xinis had ordered the administration to “facilitate” Abrego Garcia’s return, and the Supreme Court earlier this month affirmed her authority to do so.

The administration on Monday declined to answer questions posed by Abrego Garcia’s lawyers following an order by Xinis that it provide more information. Government lawyers cited attorney-client privilege, state secrets and diplomatic concerns, and told the court that questions about his whereabouts and the administration’s legal basis for his continued detention are an “absurdity.”

Xinis said in a written order Tuesday, “Defendants have failed to respond in good faith, and their refusal to do so can only be viewed as willful and intentional noncompliance.”

The judge ordered the government to provide full responses by 6 p.m. Wednesday.

Write to Mariah Timms at mariah.timms@wsj.com and James Fanelli at james.fanelli@wsj.com