The president cannot speak in front of a joint session of Congress without explicit permission from both chambers.
A resolution needs to be approved by both chambers specifying the date and time.
But Mrs Pelosi said the Democrat-controlled House would not authorise such a resolution at this time.
The traditional invitation to the president had gone out before the shutdown began.
Last week, Mrs Pelosi suggested the high-profile event be suspended due to security concerns stemming from the shutdown.
However, on Wednesday, Mr Trump wrote to the speaker formally accepting the original invitation to speak on Jan 29.
Mr Trump said he had consulted with the Secret Service and they would have “no problem” keeping him safe.
The president wrote to Mrs Pelosi: “Therefore, I will be honouring your invitation, and fulfilling my Constitutional duty to the people and Congress regarding the State of our Union.
“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!”
In a quick written response Mrs Pelosi said the House would not authorise the resolution allowing Mr Trump into the chamber.
She wrote: “I look forward to welcoming you to the House on a mutually agreeable date for this address when government has been opened.”