WASHINGTON—President Donald Trump and European Commission President Jean-Claude Juncker turned down the heat on a trade dispute between two of the world’s largest economic powers, suggesting Wednesday they would hold off on further tariffs while they talk through their differences.

Speaking in a joint news conference in the Rose Garden on Wednesday, the two leaders agreed to begin discussions on eliminating the tariffs and subsidies that hamper trade across the Atlantic, and to resolve the steel and aluminum tariffs the Trump...