Mr. Trump has since sought to blame Democrats for the shutdown, remaining in the White House in recent days rather than taking a planned extended trip to his private club, Mar-a-Lago, in Palm Beach, Fla. But he has made no effort to reach out to the incoming Democratic House speaker, Nancy Pelosi, in several days.

The president dug in further on Saturday, claiming: “For those that naïvely ask why didn’t the Republicans get approval to build the wall over the last year, it is because IN THE SENATE WE NEED 10 DEMOCRAT VOTES, and they will gives us ‘NONE’ for Border Security! Now we have to do it the hard way, with a shutdown. Too bad!”

Democrats have repeatedly indicated that they would support substantial funding increases for security at the southern border with Mexico. Many voted to do so in 2013 as part of a bipartisan, comprehensive immigration overhaul. But they view a wall like the one Mr. Trump advocates as an ineffective and costly response to an immigration system in disrepair.

Lawmakers in the party have offered Mr. Trump $1.3 billion for border security. Democratic leaders say they see little incentive to negotiate with the president after he repeatedly reversed himself on border funding, or to allocate more than $1.3 billion since the administration has spent only a small fraction of the money Congress approved last year for barriers along the border.

White House officials say that the president, who is concerned about the collapse of his own political support, feels good about his stance on the shutdown and sees no urgency toward making a deal with the incoming Democratic majority. And Mr. Trump has indicated to people that he believes the shutdown will slow down the Democrats in other pursuits, such as oversight investigations into his administration.

The children’s deaths have prompted a wave of criticism of Customs and Border Protection practices. Officials with the agency have described such occurrences as rare, and have said that the systems in place are not set up to handle the influx of families with young children trying to enter the United States.

“They were built 30 to 40 years ago for single adult males, and we need a different approach,” the commissioner of Customs and Border Protection, Kevin K. McAleenan, told CBS News last week. “We need help from Congress. We need to budget for medical care and mental health care for children in our facilities.”