Mr. Trump said “there may be the case of an Olympic champion who can get over the wall, but for the most part you are not able to do it.” He said the wall would be as high as 30 feet, roughly the equivalent of a three-story building.

In the same discussion with reporters, Mr. Trump both insisted, without evidence, that the wall was being built and could be “either renovated or brand-new by Election Day,” and reiterated his demand that Congress allocate billions of dollars for it. Mr. Trump again said that he was pushing ahead with a stretch of wall construction in Texas, claiming that he had signed a contract to build a large portion at a “great price,” but he declined to give additional details.

“Yesterday I gave out 115 miles worth of wall — 115 miles in Texas,” the president said. “And it’s going to be built hopefully rapidly. I’m going there at the end of January for the start of construction.”

Describing that section of the wall as “a big stretch,” Mr. Trump added that “we gave it out at a great price. So we’re going to have great wall there, and we have other sections to give out.”

Asked who had received the contract, Mr. Trump did not offer specifics. “Different people, different people,” he responded. “Highly bid.”

Officials with the White House and Department of Homeland Security did not respond to a request for comment on specifics about the contract.

No legislative action to resolve the shutdown is expected before Thursday, when the Senate and the House reconvene. Members of Congress have noted the increased possibility that the impasse will last at least until Jan. 3, when Representative Nancy Pelosi, Democrat of California, is expected to reclaim the speakership.