The Trump administration will continue to pursue a way of adding a citizenship question to the census, lawyers said in court filings submitted on Friday.
But administration lawyers failed to provide any legal justification for the census question by a court deadline.
The Supreme Court rejected the initial rationale for adding the question to the 2020 census as "contrived".
The question is controversial because critics believe it will discourage immigrants from taking part.
They say that lower participation by immigrants could lead to an undercount of populations in Democratic districts, benefiting President Donald Trump's Republican Party and altering how congressional seats are allocated and billions of dollars of federal funds distributed in those districts.
The Trump administration said it wanted to ask about citizenship to better enforce a law that protects the voting rights of minorities, but the Supreme Court dismissed that justification.
Why has the issue come back?
It appeared settled when government lawyers indicated they had dropped the question, and officials began printing the census without it.
That reportedly infuriated President Trump, who announced that his administration would pursue the issue.
But a deadline of 14:00 (18:00 GMT) on Friday set by a Maryland district judge came and went, with no clear indication from the administration on how they planned to add the citizenship question.
Government lawyers said only that the justice and commerce departments had been "instructed to examine whether there is a path forward".
President Trump said on Friday an executive order was among the options he was considering to force the question on to the census.
"We have four or five ways we can do it," Mr Trump told reporters, suggesting the administration could "maybe do an addendum" after getting a positive decision.
But legal experts say executive orders could not override Supreme Court decisions.
US media reported that the administration was considering using separate federal records to try to gather information about undocumented immigrants.
Why is the issue so important?
The question - "Is this person a citizen of the United States?" - has not appeared on a US census for all Americans since 1950, though it has been asked to some subsets of the population between 1970 and 2000.
The census is held every 10 years.
Democrats fear that if President Trump is successful, districts with high numbers of immigrants - such as major cities - will lose congressional representation, since census data is used to determine the distribution of federal funding and the number of congressional seats.
Congressional districts are drawn based on total populations rather than the number of legal citizens.