BRUSSELS — The United States and the European Union stepped up pressure on Monday on President Recep Tayyip Erdogan of Turkey to not use an attempted coup as an excuse for an authoritarian crackdown.

Officials even suggested that Turkey’s continued membership in NATO and its aspirations of joining the European Union could be at stake.

Meeting in Brussels, Secretary of State John Kerry and European Union foreign ministers called on Mr. Erdogan to show restraint in the wake of the failed coup, even as he continued to round up political opponents and as Turkey debated whether it should reinstate the death penalty to deal with the plotters.

“Obviously, NATO also has a requirement with respect to democracy, and NATO will indeed measure very carefully what is happening,” Mr. Kerry said. “And my hope is that Turkey is going to move in ways that do respect what they have said to me many times is the bedrock of their country.”