WASHINGTON — Speaker Nancy Pelosi pressed her case on Thursday for postponing President Trump’s State of the Union address during the prolonged partial government shutdown, even as House Democrats privately weighed action as early as next week on a homeland security spending measure that would counter the president’s demands for a wall with their own ideas for securing the border.

“The date of the State of the Union is not a special date,” Ms. Pelosi told reporters. “It’s not constitutionally required, it’s not the president’s birthday, it’s not anything. It is a date that we agreed to.”

Early Thursday afternoon, Ms. Pelosi finally received a response from Mr. Trump to the letter she sent him on Wednesday suggesting he delay his Jan. 29 speech or deliver it in writing, but he did not mention the address. Instead, the president said in his own letter that he was postponing an overseas trip Ms. Pelosi had planned with a congressional delegation, flying on military aircraft, that was scheduled to depart from Joint Base Andrews later in the day.