Trump's lawyers filed an emergency application with the Supreme Court asking the court to issue a stay, or suspension, of a grand jury subpoena demanding those tax returns and other financial records from his accountants.
The request is pending the president's plan to formally ask the high court hear his appeal of the lower court rulings that have allowed that subpoena.
If the Supreme Court agrees to hear his appeal, it will be the second time the court has taken the case, which is related to an ongoing criminal probe of the Trump Organization, the president's company, by Manhattan DA Cyrus Vance Jr.'s office.
Trump's lawyers in their filing Tuesday said "there is a reasonable probability that" the Supreme Court will take the appeal.
In the meantime, Vance has agreed to hold off on enforcing the subpoena pending the outcome of Trump's efforts at the Supreme Court.
The high court last summer rejected Trump's argument that eight years worth of his financial records from the Mazars USA accounting firm should be protected from the subpoena because of his status as president.
But the Supreme Court said Trump could make new arguments against the subpoena with a federal district court in Manhattan.
A judge in that lower court ruled against the president after Trump's lawyers argued that the subpoena was overbroad and issued in bad faith.
Vance is seeking the tax returns and other records as part of a probe that is known to be eyeing how hush money payments to women who say they had sex with Trump were accounted for by the Trump Organization. The president has denied having sex with either woman, porn star Stormy Daniels and Playboy model Karen McDougal.
Prosecutors in Vance's office also might be investigating possible tax crimes, as well as bank and insurance fraud, court filings suggest.
A spokesman for Vance declined to comment on Trump's new filing.
Trump has refused for years to release his tax returns to the public, breaking more than four decades of precedent by presidents and White House aspirants.
The president has said he will not release his returns while they are being audited by the IRS, but there is no bar on people disclosing their returns while under audit.
The New York Times last month published articles revealing that Trump paid only $750 in federal income taxes in 2016 and 2017 and no federal income taxes in other years spanning back a decade or more.
The expose also said that over the next four years, Trump faces debt repayments that he has personally guaranteed totaling more than $400 million.
-- Additional reporting by CNBC's Kevin Breuninger