Mr Trump insists the State of the Union will happen on 29 January since House Speaker Nancy Pelosi already invited him to address Congress this month.
Mrs Pelosi said that was before the shutdown dragged into day 33.
She shot back in a second letter that the lower chamber would not authorise the address before the shutdown ends.
The president told reporters on Wednesday that Mrs Pelosi had cancelled the speech “because she doesn’t want to hear the truth”.
“She’s afraid of the truth,” Mr Trump said. “She doesn’t want the American public to see what’s going on.”
Mr Trump earlier said that he was not surprised by Mrs Pelosi’s response, and said the Democrats had “become radicalised”.
“It really is a shame. This will go on for a while. Ultimately the American people will have their way because they want to see no crime, they want to see what we’re doing.”
Democrats have argued that with so many federal employees furloughed – temporarily laid off – or working without pay, the high-profile State of the Union address, which involves both chambers of Congress and the president, would not be logistically feasible.
But Mr Trump has denied that there were any security concerns posed by the ongoing partial shutdown.
What has Pelosi said ?
The Speaker, who is the third most powerful politician in the US, had invited Mr Trump at the start of January to address Congress on the agreed-upon date of 29 January.
“At that time, there was no thought that the government would still be shut down,” Mrs Pelosi said in her letter on Wednesday.
“I am writing to inform you that the House of Representatives will not consider a concurrent resolution authorising the President’s State of the Union address in the House Chamber until government has opened.”
“Again, I look forward to welcoming you to the House on a mutually agreeable date for this address when government has been opened.”
Mrs Pelosi had first asked for the State of the Union, a keynote agenda-setting speech by the president, to be delayed on 16 January, citing an unfunded Secret Service and Department of Homeland Security.
In retaliation, Mr Trump on Friday denied Mrs Pelosi use of military airplanes on a trip to Brussels and Afghanistan – less than an hour before her flight was due to depart.
He asked her to stay to negotiate an end to the partial US government shutdown.
What has Trump said ?
On 23 January, Mr Trump dug his heels in over the annual address, telling Mrs Pelosi: “It would be so very sad for our Country if the State of the Union were not delivered on time, on schedule, and very importantly, on location!”
He emphasised that Mrs Pelosi had already invited him to speak, he had accepted that invitation, and that security agencies informed him “there would be absolutely no problem regarding security with respect to the event”.
“Therefore, I will be honouring your invitation, and fulfilling my Constitutional duty,” he added.
Why is the US government partially shut down ?
A row between the Republican president and Democrats over border security has led to the longest government shutdown in US history.
Mr Trump is demanding $5.7bn (£4.4bn) of congressional funding to build a wall on the US-Mexico border, but newly empowered Democrats have refused.
Some 800,000 federal employees have been going unpaid since 22 December as a result of the shutdown.
Donald Trump says he wants to deliver his State of the Union speech “on time, on schedule and on location”. He can accomplish the first two without difficulty.
The Constitution is silent on the how the president “from time to time” must communicate with Congress.
The “location”, however, could be tricky.
As the president must know, House Speaker Nancy Pelosi has near total control over her chamber of Congress.
While she did invite the president to speak earlier this month, the legislative action to allow the president to give his joint address has yet to be implemented.
Mrs Pelosi has said she’s not budging until the government reopens – and in this particular case, her word is final.
All this sets up a game of chicken over a speech within the game of chicken over the shutdown. Everything up to this point has been manoeuvring to gain tactical advantage and avoid the public’s scorn.
For the moment, polls show Americans holding the president responsible for the impasse. But the fallout from this unprecedented State of the Union drama, if it reaches a climax on Tuesday, is unpredictable, to say the least.