(CNN) As US officials scramble to track down hundreds of parents who were deported without their children, attorneys are sparring over a key question: Should deported parents be brought back to the US as part of reunification efforts?

Government attorneys argue that's taking things too far. But ACLU attorneys say it could be the only fair option for some parents, who might have given up valid asylum claims thinking that would lead to swifter reunifications with their children.

"There may be specific individuals who were misled and coerced," the ACLU's Lee Gelernt said in a phone hearing Friday.

It's the next legal battle in the high-profile federal lawsuit over immigrant families the US government separated . And it could impact hundreds of deported parents. According to the latest government statistics, there are 366 children from separated families in custody whose parents are now outside the United States.

In a court filing this week, Justice Department attorney Sarah Fabian argued the government is not required to bring any deported parents back to the United States, and that any order to do so would go against what the court has said is the government's right to enforce immigration laws and exercise discretion.

US District Judge Dana Sabraw didn't rule on the matter Friday, instead asking attorneys from both sides to confer and provide more details about the issue in a filing next week.

But the judge shared what he said were his "tentative thoughts" on the matter so far, noting that the government's plan to reunite children with deported parents in countries of origin is appropriate.

Sabraw said he sees many issues with the ACLU's proposal that some deported parents should have the chance to be flown back to the United States to pursue asylum requests jointly with their children.

"That would certainly be a significant undertaking, and would be over the strong objection of the government. And I'm not sure about all the jurisdictional arguments that may be raised," Sabraw said. "And then, from a practical standpoint, it seems to me is that in the best interests of the family, what we're looking for is reunification, and that under the pressing circumstances, that reunification ought to occur in the home country."

Sabraw said he "approved wholeheartedly" of the latest reunification plan, which outlines steps that US agencies and an ACLU-led steering committee will follow as they work to track down deported parents and continue efforts to reunite families the government separated.

"What I'm expecting is that everyone's moving full speed ahead on every possible angle," he said.

Making contact

The ACLU says it's having trouble reaching deported parents, even after getting phone numbers from the government.

"We have been making calls and we have called a good many ... around a third of them. We are unfortunately not reaching many people," Gelernt said in Friday's hearing. "It seems the phone numbers may be inoperative and some people may be in hiding, so I think we're going to be circling back with the government to check those numbers."

Some people reached by the ACLU and its partners say they want to be reunified with their children in their home country, Gelernt said. Others say if the option is available, they'd like to return to the United States and seek asylum.

Gelernt said he believes the ACLU has called about 120 people so far.

"But I believe we've only reached a little less than 50," he said. "So that's been a problem."

Sabraw asked attorneys for the ACLU and the government to provide more details next week on the status of efforts to contact deported parents, and efforts to reach parents who were released from custody in the United States.

He also asked them to provide proposed next steps for how children who were reunited with their parents in the United States can continue to pursue asylum claims.

The judge issued a temporary restraining order on Thursday, blocking the government from deporting any reunified families. Children who've been reunified with their parents, Sabraw said, should still have the chance to seek asylum in the US.

"This is an enormous undertaking involving a situation of the government's own making. We will never be able to come up with a process that is perfect or that restores all rights as if this incident never happened," Sabraw said. "All we can do is the best we can do under the present circumstances."