WASHINGTON—Senate Republicans threatened Tuesday to block the White House’s planned tariffs on Mexico, hours after President Trump signaled he was prepared to move ahead with the levies barring a last-minute deal over border security.

Mr. Trump, frustrated with the flow of asylum-seeking Central American families at the southern U.S. border, said last week that the U.S. would impose escalating tariffs on all Mexican imports starting on Monday unless Mexico took sufficient steps to stop migration. U.S. and Mexican negotiators...