WASHINGTON — John F. Kelly, the homeland security secretary, said Wednesday that it was doubtful that a wall along the full border with Mexico would ever be built, despite an oft-repeated campaign promise by President Trump.

“It is unlikely that we will build a wall from sea to shining sea,” Mr. Kelly told senators on the Homeland Security Committee.

Instead, Mr. Kelly said, the department will look to build physical barriers — including fencing and concrete walls — in places that make sense. He said that the department was still studying the best places to construct such barriers, and that he could not give an estimate of the cost.

The first bids for prototypes of the border wall were due Tuesday. According to people briefed on the agency’s plan, the first new section of the wall will be built on a short strip of federally owned land in San Diego, where there is already fencing.

Congress has not yet acted on the funding request for the wall, and it faces considerable opposition from Democrats and even some Republicans.

Speaker Paul D. Ryan said Congress probably would not grant the Trump administration’s request for additional money for the border wall this year, adding that it would most likely be included in the next fiscal year’s appropriation.

In his budget released this month, Mr. Trump asked for $1.4 billion to pay for the initial development of the wall. The Department of Homeland Security said enough funding had been moved from other programs to begin construction.

Democrats have vowed to block Mr. Trump’s budget proposals to build the wall and add Border Patrol and deportation agents.

Mr. Kelly’s department has begun implementing key features of Mr. Trump’s executive orders to increase border security and crack down on illegal immigration. Federal judges have blocked the department from carrying out the president’s executive orders seeking to ban travel to the United States by citizens of six mostly Muslim countries.

Nevertheless, Mr. Kelly said the department was proceeding with other efforts to strengthen border security and was starting to see some results. For example, he noted that apprehensions along the southwestern border had declined. Border Patrol agents caught almost 17,000 people trying to cross the border illegally last month, down from nearly 60,000 people in December.

Mr. Kelly said stepped-up enforcement had forced smugglers to raise their prices, which most likely contributed to the decline in crossings.

Under questioning by senators on Wednesday, Mr. Kelly defended several Trump administration policies, including what the Homeland Security Department calls “extreme vetting” of international travelers, the searching of electronic devices at the border, and the possible separation of mothers and children at the border to discourage immigration.

Mr. Kelly told senators that children would not be routinely separated from their families unless the “situation at the time requires it,” such as when the mother is sick or addicted to drugs.

Senator Claire McCaskill of Missouri, the ranking Democrat on the committee, raised concerns that the Homeland Security Department and the State Department were considering asking visitors applying for a visa to the United States for access to their cellphones, financial records and passwords to social media accounts.

Ms. McCaskill called the proposal “un-American” and said it could set off a diplomatic row with other countries.

“I’m worried that if we apply ‘extreme vetting’ procedures like this, American travelers will be forced to undergo the same scrutiny if these countries decide to reciprocate,” she said. “We are doing things that in no way trips up the bad guys.”

Mr. Kelly said that the visa-vetting procedures built on polices that had been in place for years and that foreign visitors could choose not to share their personal details.

“If they don’t cooperate, they can go back,” he said.

Asked about searching electronic devices at the border, Mr. Kelly said that just a small number of people had had their phones or computers searched. And he said the searches had yielded results, catching pedophiles and other criminals.

But that failed to persuade Senator Rand Paul, Republican of Kentucky, who called the policy “obscene” because it also affected citizens and green card holders.

“There is a difference between searching my bags and searching my phones,” said Mr. Paul, who added that he was worried that customs agents were downloading contacts and other information from cellphones.

Mr. Kelly got into a heated exchange with Senator Kamala Harris, Democrat of California, who pressed him to put his policy on separating families into writing for the sake of border agents. Mr. Kelly said he had done so orally and did not need to put it in writing.

Ms. Harris also questioned the use of more intense enforcement efforts to round up undocumented immigrants, including by sending agents from Immigration and Customs Enforcement into courthouses to arrest witnesses and victims.

“They are now allowed to do their jobs,” Mr. Kelly replied.