Speaker of the House Nancy Pelosi said in a letter to Mr Trump that "unless government re-opens this week" the event should be re-scheduled.
Mr Trump is due to address Congress for the annual State of the Union speech on 29 January.
The US government is partially shut shut to a row over border wall funding.
In Wednesday's letter, Mrs Pelosi cited the "extraordinary demands presented" by the event as the reason for postponing.
She wrote: "Both the US Secret Service and the Department of Homeland Security have not been funded for 26 days now - with critical departments hamstrung by furloughs.
"Sadly, given the current security concerns and unless government re-opens this week, I suggest we work together to determine another suitable date after government has re-opened for this address or for you to consider delivering your State of the Union address in writing to Congress on January 29th."
Has this speech ever been delivered in writing before?
The US constitution requires that the president "shall from time to time give to the Congress Information of the State of the Union".
According to the US National Archives, both the first US President, George Washington, and his successor, John Adams, delivered their speeches in person.
It was the third US President, Thomas Jefferson, who began the practice of delivering his State of the Union addresses in writing.
This was the tradition for more than a century until President Woodrow Wilson spoke in person to a joint session of Congress in 1913.
With rare exceptions, all subsequent US presidents have opted to appear in person on Capitol Hill every year to deliver this message setting out their administration's aims and achievements.
The event is attended by every member of the US Senate and House of Representatives, justices from the Supreme Court and military leaders.
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