SINTRA, Portugal—European Central Bank President Mario Draghi signaled the bank could roll out fresh stimulus as soon as July, sending the euro lower against the dollar and prompting an unusual rebuke from President Trump.

Mr. Draghi said Tuesday that ECB policy makers would consider in the coming weeks how to adapt their policy tools “commensurate to the severity of the risk” to the economic outlook. Options include extending the time frame before the next interest-rate increase, a reduction in the already negative policy...