WASHINGTON — Dozens of U.S. citizens stranded by the fighting in Yemen have asked the Obama administration to help them flee the country, according to a lawsuit filed in federal court seeking to compel an American rescue operation.

The suit was filed as Houthi militants appear to be gaining ground against the government’s security forces, and as Saudi-led airstrikes against the Houthis have halted most air travel and blocked seaports.

Russia, India, China and five other countries helped their citizens leave Yemen in recent weeks. The State Department and the Pentagon flew U.S. diplomats and special-operations troops from the country this spring, but did not evacuate other U.S. citizens, and now say doing so would be too risky.

Advocacy groups say at least 500 U.S. citizens or Yemenis with permanent-residence status in the U.S. are caught in the country.

On Thursday, advocates for 41 Yemeni Americans seeking to get out filed a lawsuit in U.S. District Court for the District of Columbia demanding that Secretary of State John Kerry and Defense Secretary Ashton Carter “use all resources at their disposal” to evacuate them and other U.S. citizens from Yemen.

For the U.S. not to evacuate its citizens from Yemen, when it has done so in other conflicts, is “arbitrary and capricious,” they said.

State Department spokesman Jeff Rathke declined to respond to the lawsuit. He said the State Department repeatedly advised Americans in Yemen to leave the country before the fighting worsened.

“For more than 15 years, the State Department has been advising U.S. citizens to defer travel to Yemen, and we’ve been advising those U.S. citizens who are in Yemen to depart,” Rathke said at the State Department.

Instead of waiting for a U.S. evacuation, Americans in Yemen should try to board flights out of the capital, Sanaa, arranged by the International Organization for Migration, a Geneva-based organization that assists refugees, Rathke said.

The U.S. vacated its embassy in Sanaa in February and pulled its remaining special-operations teams and intelligence officers from a Yemeni air base last month.

“We are, you know, unfortunately, in a situation where access to Yemen is extremely difficult and to do so with U.S. government assets could put other lives at risk,” Rathke said.

Last week, the State Department urged Americans in Yemen to board ships and planes that the governments of India and Djibouti were using to ferry their citizens to safety. Those evacuations ended Friday.

Since late March, Indian airplanes and naval vessels have removed 4,640 Indians and 960 others, according to India’s Ministry of External Affairs.

Yemen has become increasingly cut off, with anti-government militias at land border crossings, and airports and seaports under attack. Fuel for cars is scarce and expensive, and travel by road is dangerous.

Abed Ayoub, legal and policy director of the American Arab Anti-Discrimination Committee, one of the advocacy groups behind the lawsuit, said in a telephone interview: “The U.S. has the strongest military in the world. We shouldn’t have to rely on other countries to get our citizens out.”

During the Lebanon-Israel conflict in 2006, a similar lawsuit prodded the State Department to arrange a ship to take U.S. citizens from Lebanon to safety on the island of Cyprus, Ayoub said.

“We are looking for the State Department to step up their effort to get them out of harm’s way,” he said.

Hesham Hussain, 30, a sales manager in Santa Clara, Calif., said he wants the U.S. to help evacuate his father, Abdu, a U.S. citizen. The older man retired to Taizz, Yemen’s third-largest city, in 1992 after he sold the family grocery in Oakland.

“If the Indians and Chinese are able to do it, I have no doubt the U.S. can do it,” Hussain said on the phone from Santa Clara.