In a bid to repair a rift with Mexican President Enrique Peña Nieto, President Donald Trump agreed on Friday to stop speaking publicly about one of his signature campaign promises: that Mexico would pay for the construction of a wall along the U.S. border.

The concession was detailed in the joint U.S.-Mexico statement after the two leaders had an hour-long phone call, but not in the White House’s version. A White House official said Mr. Trump did indeed agree to stop talking publicly about Mexico paying for the wall, which...