If the court were to grant the extension, it could mean that justices might only hear the case next term, and possibly only decide it after the presidential election.
Patrick Rodenbush, a Justice Department spokesman, said the department intends to oppose Texas' request.
"The case presents issues of national importance and the department believes it should be considered expeditiously," he said.
It was only last week that lawyers for the Justice Department asked the court to reverse a lower court ruling and allow President Barack Obama's controversial immigration programs, meant to ease deportation threats to millions of undocumented immigrants, to take effect. The programs target eligible parents of U.S. citizens as well as some individuals who entered the country as children.
In the letter to the court, Texas Attorney General Ken Paxton, writing on behalf of the 26 states that are challenging the programs, argues that there "is good cause" for the extension "in light of pressing deadlines in other cases handled by counsel for the State respondents."
The clerk of the court ordinarily handles extensions, says John Elwood, a lawyer who argues frequently before the court. If the extension request is opposed, the request would likely go to Justice Antonin Scalia who has jurisdiction of the 5th U.S. Circuit Court of Appeals. Elwood says that even if the extension is granted, it is not out of the question that the court could decide the case before the next election.
"Even if the case is delayed, there is a chance the court could grant expedited briefing, or a special extra sitting (as it did in the Citizens United case) but those options would be unusual," said Elwood. "Timing will be very tight.'