Ms Pelosi said the president will not be allowed to deliver the address to a joint session of Congress on Tuesday, in spite of his suggestion he would show regardless of whether he was welcome.
The House will not approve a resolution allowing Mr Trump to come until the government reopens, Ms Pelosi said.
Donald Trump responded by saying “we’ll do something in the alternative”, while branding Democrats “radicalized.”
The argument comes shortly after a new poll revealed Donald Trump’s disapproval rating has risen to 57 per cent - the highest recorded by the pollster since his presidency began. The Politico / Morning Consult survey found that a majority of Americans blame the president for the government shutdown.
With thousands of furloughed workers still not receiving pay, tensions are running high over the ongoing closure.
Police were called to a protest by a group of federal government workers at a field office for Senator Mitch McConnell in Kentucky.
The Senate Majority Leader blocked legislation on Wednesday that would reopen most of the government.
Some 800,000 federal workers are set to miss a second pay check this Friday.
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