Its prospects looked dim. The Democratic-controlled U.S. House of Representatives has passed similar bills but Trump has rejected legislation that does not include the wall funding.
The mere fact that McConnell, who previously said he would not consider legislation that Trump did not support, is willing to allow for a vote suggests he may be trying to persuade lawmakers of both parties to compromise.
He also planned to hold a vote on a bill including wall funding and a temporary extension of protections for "Dreamers," hundreds of thousands of people brought to the United States illegally as children, an offer Trump made on Saturday.
Democrats have dismissed Trump's offer, saying they would not negotiate on border security before reopening the government and would not trade a temporary extension of the immigrants' protections in return for a permanent border wall they have called ineffective, costly and immoral.
McConnell's calculation may be that if both bills fail, that might help convince Republicans and Democrats to look for a compromise.
One possibility emerged on Wednesday when House Democratic leaders floated the idea of giving Trump most or all of the money he seeks for security along the Mexican border but that could not be used to build a wall.
Representative James Clyburn, the No. 3 House Democrat, said Democrats could fulfill Trump's request for $5.7 billion for border security with technological tools such as drones, X-rays and sensors, as well as more border patrol agents.
On Wednesday, House Speaker Nancy Pelosi, the top U.S. Democrat, essentially disinvited Trump from delivering the annual State of the Union address in the House chamber until the government is fully open. Trump called her move "a disgrace."
A Reuters/Ipsos poll last week found more than half of Americans blamed Trump for the shutdown even as he has sought to shift blame to Democrats after saying last month he would be "proud" to close the government for border security.