Trump's veto sends the resolution back to the US House of Representatives, which is expected to pick it up after the week-long congressional recess.
Lee proposed a version of the resolution that would have restricted future emergency declarations while grandfathering in Trump's, but he couldn't reach a compromise agreement with the White House, and the plan fizzled.
Before the Senate voted on terminating Trump's national emergency, he told reporters, "I'll probably have to veto it", and predicted his veto wouldn't be overturned by Congress.
The Senate voted Thursday by a margin of 59 to 41 to block Trump's declaration that an emergency on the border with Mexico requires the construction of a wall, a project for which Congress has not appropriated money.More news: Wake headed to Titans
Trump wants to use the emergency order to divert billions of federal dollars earmarked for defence spending toward the southern border wall.
Trump had sought to frame the issue as voting for him or for "Nancy Pelosi, Crime and the Open Border Democrats".
The president tweets that he looks forward to vetoing the Senate resolution that would increase "crime, drugs and trafficking"; Kevin Corke reports from the White House.
Earlier today, President Trump thanked Republicans who voted against the measure. "Watch, when you get back to your State, they will LOVE you more than ever before!" he said.More news: Chiefs agree to deal with DE Okafor
The other Republicans who voted to oppose the declaration were Sens.
Alexander added that he shares the president's goal of border security, but warned allowing Trump to redirect money without authorization could set a risky precedent.
The emergency declaration is being challenged in court as an unconstitutional usurpation of Congress' power of the purse. The large numbers have put a strain on immigration enforcement, although critics say a wall would do little to discourage the crossings since many of those apprehended at the border turn themselves in voluntarily.
"The legal scholars all say it's totally constitutional", Trump said during a photo op Thursday in the Oval Office. Attorney General William Barr is also at the President's veto event.More news: Hammond talks up United Kingdom economy despite Brexit 'cloud'