If the outgoing president's plan goes forward, states with large numbers of undocumented immigrants could see their influence reduced in the US House of Representatives.
It would amount to a last-minute victory or defeat for Trump, who is due to leave the White House and hand over to President-elect Joe Biden on January 20 even though he is still refusing to concede his electoral loss.
The US census is carried out every 10 years as spelled out in the Constitution, and it determines certain federal aid and the number of seats each state holds in the House of Representatives, the lower house of Congress.
In July, with the census underway, Trump issued a directive to exclude the country's estimated 10 million undocumented immigrants from the count determining the number of House seats.
The Republican who has spent his presidency seeking to limit immigration has said he does not want to allow congressional representation to foreigners in the country illegally.
Until now, the census has included all residents of a state apart from foreigners on a temporary visa.
Several Democrat-led states, including New York, which has a large number of immigrants, have challenged the change and have been victorious in lower courts.
The Trump administration as a result asked the Supreme Court to intervene urgently since the president is due to transmit the results of the 2020 census and the number of seats for each state to Congress in January.
Monday's arguments before the court will be by telephone due to the Covid-19 pandemic, and a decision is expected soon.
In 2019, a narrow majority on the court blocked an attempt to ask for the citizenship of census participants, which risked further reducing foreigners' willingness to respond and undercounting the population in certain states.
Since then, another Trump appointee has been named to the Supreme Court, with conservative justices now holding a six-to-three majority.
The justices could however limit themselves to questions of procedure in the case and avoid the issue entirely.
According to a study from the Pew Research Center, three states - California, Florida and Texas - could lose a seat each over the next decade if the change goes forward.
Three others - Minnesota, Alabama and Ohio - could gain one seat, Pew said.
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