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.