WASHINGTON — As a Friday night deadline approaches to prevent a government shutdown, one hot-button spending issue is a down payment to build a wall on the southern border, one of President Trump’s signature national security goals.

Mr. Trump appeared to soften his demand late on Monday, saying he would settle for additional funds for border security. But then on Tuesday, he insisted again, “The wall’s going to get built, folks.”

In the run-up to this divisive debate, John F. Kelly, the homeland security secretary, has escalated his rhetoric about threats facing the United States.

“The risk is as threatening today as it was that September morning almost 16 years ago,” Mr. Kelly, a retired four-star Marine Corps general, said in a speech last week at George Washington University. Mr. Kelly suggested that undocumented immigrants may not be the only threat coming across the border with Mexico, pointing to the possibility of terrorists using the same smuggling routes to infiltrate the United States.