Tuesday's request signalled that the Biden administration is likely to scrap the rule on its own, Xinhua news agency quoted a report published by The Hill news website.
The report noted that the White House has already mandated a review of the rule by the Department of Homeland Security in a February executive order.
The highest court agreed last month to hear challenges to the 2018 rule by a number of states and advocacy groups that argued it amounted to a wealth test for immigrants.
The Trump administration insisted it was in the best interests of the US to ensure immigrants could be self-sufficient.
"Immigrant families can now access life-saving health care, food, and housing assistance for which they are eligible without fear that they will lose the chance to obtain lawful permanent residence, because the actions today mean that the harmful Trump public charge rule will again be blocked," the Legal Aid Society, Catholic Legal Immigration Network Inc. and others who had challenged the rule said in a joint statement on Tuesday.
"The Trump rule erected an invisible wall in the form of a wealth test that discriminated against people on the basis of race as a condition for regularising their immigration status.
"And because of the public charge rule, immigrant families have been living in fear of using essential benefits like healthcare, despite serving as frontline workers who have been among those hardest hit by Covid-19," it added.
The Biden administration has taken a similar posture in other major immigration cases, convincing the Supreme Court to cancel hearings on cases challenging Trump's border wall and his changes to the asylum process, according to the report.