“So unless the wall is 35,000 feet high, it's not going to do much to stop those overstaying these visas,” said Robert Warren, a fellow at the Center for Migration Studies who has worked on immigration issues for Republican and Democratic presidents.

Additionally, Mr. Warren said, many people who have been stopped by the Border Patrol in recent years are seeking asylum — including some who simply walk up to agents and surrender.

Mr. Trump’s budget request for a wall represents more than half of the $33 billion spending blueprint for border security over the next decade. It either eliminates critical funding for border security programs or shifts money from them, threatening to leave gaping holes. A Government Accountability Office study released last February found that Customs and Border Protection has not shown how much fencing and walls bolster border security.

An internal budget guidance document for the 2019 fiscal year shows that the White House Office of Management and Budget asked officials at the Homeland Security Department to reduce or delay funding requests for additional border security technology and equipment. Instead, the document instructed, Homeland Security should dramatically increase funding for a wall on the Mexico border.

Homeland Security officials said the plans are subject to change. Still, the document underscores the priority that a border wall remains for Mr. Trump, who promised its construction during his presidential campaign. It also instructed the department to seek $1.6 billion in the upcoming fiscal year to build 74 miles of a border wall — about $700 million more than Homeland Security officials felt they needed to build that.

Parts of the document were viewed by The New York Times; the rest of it was based on reports by the Democratic staff of the Senate Homeland Security Committee.