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Facing serious overcrowding in immigration courts, federal authorities in New York adopted a high-tech solution last year: Immigrants would be kept in detention centers for their legal proceedings, appearing before judges only by videoconference.

Now, a new lawsuit claims that the policy infringes upon immigrants’ constitutional rights in a deliberate attempt to speed up and increase deportations.

The lawsuit, filed on Tuesday in the Federal District Court in Manhattan, asserted that detained immigrants could not fully communicate with their lawyers and participate in proceedings when their only interaction with immigration court was through video.

As a result, the lawsuit said, immigrants who might otherwise be granted the ability to stay in the United States instead could be deported. The suit cited several instances when videoconferencing had harmful effects on immigrants and their hearings.

One immigrant spent three extra months in detention because his jail’s one videoconferencing line connecting inmates to immigration court had been double booked.

Another was afraid to testify about his sexual orientation because Immigrations and Customs Enforcement officers at the detention facility where he was being held were listening as he spoke to the court in a videocall.

The interpreter for a third had difficulty understanding an immigrant as he sobbed through his two-hour testimony, because his words were muffled over the audio feed of the videoconference.

ICE has justified videoconferencing as an efficient way to address the backlog at the immigration court on Varick Street in Manhattan.

An ICE spokeswoman said she could not comment on pending litigation.

But the lawsuit said the videoconferencing was “merely pretext for the true reasoning behind the policy — limiting due process, access to the courts and counsel for immigrants in an effort to rush deportations and deport more people.”

Immigration courts have been overwhelmed by cases for over a decade, but the suit put a spotlight on efforts under the Trump administration to tackle the backlog by speeding up proceedings. The strategy has drawn criticism from immigration lawyers, advocates and judges, who argue that expediting proceedings can rob immigrants of a fair trial.

To address overwhelmed dockets, former Attorney General Jeff Sessions announced in April 2017 that the Department of Justice would bring on an additional 125 immigration judges, at least 44 of whom have been hired.

Mr. Sessions also imposed policy changes the Justice Department called common-sense reforms to increase the efficiency of immigration courts.

But immigration lawyers argued that prioritizing speed has come at the expense of due process.

“This push toward efficiency over due process is particularly Trumpian,” Andrea Sáenz, the supervising attorney at Brooklyn Defender Services, said. The videoconferencing policy in New York is part of a larger trend of addressing the backlog by focusing “on what the government or ICE thinks is faster or cheaper, instead of giving people full and fair hearings,” Ms. Sáenz said.

The plaintiffs in the federal suit include Brooklyn Defender Services, the Legal Aid Society and the Bronx Defenders, all of which offer free legal services to immigrants facing deportation as a part of the New York Immigrant Family Unity Project, a program funded by the New York City Council. Low-income people in immigration detention do not otherwise have a right to a lawyer if they cannot afford one.

In previous years, judges frequently used their discretion to close some cases they deemed “low priority,” to take them off their dockets indefinitely and reduce the backlog.

But Mr. Sessions restricted judges’ ability to do so, arguing in a decision in September that closing certain cases was “not a free-floating power an immigration judge may invoke” solely to create space on their docket. Advocates said this has added to the backlog.