WASHINGTON—Top lawmakers from both parties emerged from a meeting Tuesday with President Barack Obama saying they will support military action in Syria to deter the future use of chemical weapons, pushing the U.S. closer to military strikes against the regime of President Bashar al-Assad.

House Speaker John Boehner (R., Ohio) said after the meeting that only the U.S. has the capability to stop Mr. Assad and warn others about the use of chemical weapons.

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