MEXICO CITY — Amid the diplomatic showdown over President Trump’s order to build a border wall — and who should pay for it — Mr. Trump and President Enrique Peña Nieto of Mexico spoke by telephone for an hour on Friday and agreed to proceed with negotiations on a range of bilateral issues, both leaders said.

The presidents made an effort to say that the call had been productive, implying that tensions had cooled, but they stood their ground on their positions, with no suggestion that the conversation had resolved their disagreement over fundamental issues, most notably the payment for construction of a wall between the United States and Mexico.

“With respect to payment for the border wall, both presidents recognized their clear and very public differences in their stances on this very sensitive issue, and agreed to solve these differences as part of a comprehensive discussion of all aspects of the bilateral relationship,” according to a statement issued by Mr. Peña Nieto’s office.

The statement added: “The presidents also agreed for now not to speak publicly about this controversial issue.”