Mr. Trump did not say what his options were. But people familiar with the discussions said that among the options being considered was using federal records other than the census questionnaires sent to every household to try to glean information about undocumented immigrants. Census Bureau officials have said that method would produce data on citizenship better and more cheaply than a citizenship question, and Commerce Secretary Wilbur Ross, who oversees the Census Bureau, ordered that data collected when he approved the citizenship question in March 2018.

Another possible option is an executive order, but it was not clear what form it would take or what it would accomplish in light of last month’s Supreme Court decision rejecting the justification Mr. Ross had given for adding the citizenship question.

But the Constitution assigns the responsibility for overseeing a decennial census to Congress, not the president. And Congress has limited executive authority over the census, as the Supreme Court recognized.

“The taking of the census is not one of those areas traditionally committed to agency discretion,” Chief Justice Roberts wrote in the opinion on the citizenship question, meaning that it cannot be accomplished by unilateral executive action.

An executive order from the president could be an attempt to speed the process to allow expedited court review of a new justification. Or it could be an attempt to assert that no justification is needed beyond executive authority.

In their filing on Wednesday in Federal District Court in Maryland, Justice Department lawyers argued that a new rationale for the citizenship question should be assessed on its own merits, without reference to the first attempt that the Supreme Court blocked. They cited last year’s Supreme Court decision sustaining Mr. Trump’s executive order limiting travel from several predominantly Muslim countries, noting that the justices had considered only the last of three orders on the matter.

Late Friday, Judge Hazel rejected that argument when he allowed the case to go forward.

Opponents of the census question attacked it on another front late Friday, telling the federal judge in a Manhattan lawsuit that the government’s effort to resurrect it flew in the face of its repeated insistence that the printing of census forms could not be delayed past June 30.