That would give Republicans who are uneasy about the constitutionality of the February 15 declaration - yet nervous about publicly rebuking Trump - some political cover to side with the president.
Barring a last-minute deal on some other face-saving measure, up to a dozen Republicans could join all Senate Democrats in voting to send the disapproval resolution to Trump's desk, setting up the first veto of his presidency.
President Donald Trump speaks during an event in the Rose Garden at the White House to declare a national emergency in order to build a wall along the southern border, February 15, 2019, in Washington.
However, the bill would need Trump's signature to become law, and Lee ostensibly hoped to exchange his support for the border declaration for Trump's promise to approve the bill limiting his powers. The official was not authorized to publicly private conversations and spoke on condition of anonymity.
At stake is billions of dollars in funding for a wall along the U.S. -Mexico border that Trump is demanding but Congress has refused to fully provide.
The resolutions are both privileged, which means they can not be stopped by Senate Majority Leader Mitch McConnell, a Republican from Kentucky.
Trump confirmed his veto threat for the national emergency resolution in comments from the White House on Wednesday afternoon. The bill would also give Congress the power to limit or amend the scope of any emergency declaration, and require the president to report how exactly his emergency powers are being put to use.
The disapproval resolution has already passed the House.
Other Republican senators who back the resolution are Alaska's Lisa Murkowski, Maine's Susan Collins and Kentucky's Rand Paul.
The votes on the Yemen resolution and the emergency declaration represent some of the strongest backlash the President has weathered from his own party yet. Tillis faces a potentially tough re-election fight next year. Lisa Murkowski of Alaska, Susan Collins of ME and Rand Paul of Kentucky have all publicly announced their support for the resolution of disapproval, with more Republicans privately wrestling with their votes.
"I'm voting with you, Mr. President", Oklahoma Republican Sen.
"Every single Democrat running for president should be made to answer: do they agree with the Speaker who stands in opposition to baseless impeachment charges, or will they risk fracturing the country by bowing to the radical elements in their party who want to disenfranchise the American people and overturn the legitimate and lawful result of the last election?"
In the meantime, House Oversight Chair Elijah Cummings of Maryland is locking horns with an increasingly recalcitrant White House over issues related to Trump's hush money payment to Stormy Daniels and the top-secret security clearances granted to Trump family members and White House advisers Jared Kushner and Ivanka Trump. But aides expected that the vote on the resolution of disapproval on the national emergency declaration could peel off anywhere from 10 to 15 Republicans, a sizable chunk.
That 30-day approval window would apply after Trump's current border declaration expires in one year, meaning it could remain in place for at least one year without congressional approval.
Congress would be highly unlikely to muster the two-thirds majorities needed to eventually override a veto.