The measure, pushed through Friday by House Republicans, would fund the government after Sept. 30 if all spending for President Obama's healthcare law was eliminated.
Sen. Ted Cruz, R-Texas, has been pushing for a filibuster on the bill because it would require that Democrats have a supermajority of 60 votes to strip any provision to defund Obama's healthcare law from the legislation.
McConnell says he supports the House Republicans' bill and will not vote to block it, NBC reported. A spokesman said the Kentucky Republican will also vote against any amendment that attempts to add funding for the healthcare law back into the bill.
Sen. John Cornyn, R-Texas, said he also supports the measure. "I intend to support the House bill that defunds Obamacare and will vote against a bill that funds it," the minority whip said Monday in a tweet.
The White House warned Monday a government shutdown would inflict an unnecessary wound on the U.S. economy.
"Everybody gets hurt when Washington does something stupid like inflict a wound on the economy unnecessarily, in which, in this case, Republicans would be doing it," spokesman Jay Carney told reporters Monday en route to New York aboard Air Force One.
"Default would invariably lead to a global financial and economic crisis. And just toying with the prospect of it, as some Republicans continue to do, is harmful."
Vice President Joe Biden promised Monday Federal Emergency Management resource centers and hotlines will remain open to assist Colorado flood victims regardless of what happens in Washington.
Biden visited a makeshift FEMA resource center in Greeley, Colo., Monday afternoon after surveying flood-damaged communities by helicopter.
"It's probably going to scare the living devil out of you," Biden said of the negotiations over the debt ceiling that threaten to shutdown the federal government. But, he said, "there will be someone on the other end of the line who will walk you through."
The Pentagon says U.S. military service members will remain on duty and work without pay if the government is forced to shut down.
Pentagon spokesman George Little said service members would be paid in full after the shutdown ends, The Hill reported Monday.
Civilian Pentagon employees, however, would not receive retroactive pay, he said.