If an appeal is not filed by the deadline, the stay will dissolve.
In March, the District of Columbia Circuit Court of Appeals issued a ruling in a separation-of-powers lawsuit brought by House Democrats clearing the way for the documents to be released.
The appeals court gave the Justice Department until May 11 to turn over the materials to Congress, but the Supreme Court on May 8 issued a stay to allow both sides to submit legal briefs.
On Tuesday, House Democrats filed a brief with the Supreme Court to gain access to the documents, while also suggesting its investigation into potentially impeachable offenses has expanded to include Trump allies Roger Stone and Michael Flynn.
"If this material reveals new evidence supporting the conclusion that President Trump committed impeachable offenses the committee will proceed accordingly -- including, if necessary, by considering whether to recommend new articles of impeachment," House general counsel Douglas Letter said.
Letter also said he was concerned that it could take the Supreme Court months to issue a decision in the case if it decides to take it up, potentially hindering the House's ability to complete the investigation.
Senate Republican leader Mitch McConnell on Wednesday pointed to the court decision, stating Democrats are attempting to conduct a "perpetual investigation, perpetual impeachment" of Trump.