Challengers to the ban, who say the order is meant to discriminate against Muslims, have said in court papers that it should expire on Wednesday, 90 days after the executive order was due to take effect on 16 March.
Trump was able to make the appointment of Gorsuch only because the Republican-led Senate past year refused to consider former President Barack Obama's nominee to the court, Merrick Garland.
An entry on the Supreme Court docket said that the administration can file its new brief on Thursday.
On Wednesday, the Trump administration modified the start date of the proposed travel ban executive order in an attempt to keep it alive, Reuters reported.
The Trump administration is asking the Supreme Court for additional time to file legal papers in response to a second federal appeals court that refused to lift a hold on President Donald Trump's travel ban.
"Together with my brother bishops, we believe it is possible to simultaneously provide for the security of our country and have a humane refugee policy that upholds our national heritage and moral responsibility".
In appointing Gorsuch, Trump restored the court's 5-4 conservative majority.
The ruling given on Monday against the travel ban was yet another block against Trump's efforts to prohibit entry of citizens of Iran, Libya, Somalia, Sudan, Syria and Yemen for 90 days while the U.S. government reviews their screening procedures. That ruling said the executive order violated US immigration law. The Supreme Court is weighing emergency applications in both cases, but is likely to act on them together. Because he issued the ban back in March, those 90 days are up this week.
The Supreme Court on Tuesday set a new briefing schedule, meaning the highest court in the judicial branch in the coming days will decide whether to lift the temporary lower court injunction and whether to hear the Trump administration's appeal against the injunction that blocked the order's implementation.