Later this week, the Senate will vote on a continued spending resolution to keep the government funded.
The government would shut down Oct. 1 if Congress doesn't approve some form of government funding measure.
McConnell and other Republicans are trying to encourage vulnerable Democrats up for reelection in red states to vote against funding ObamaCare as a way to prevent implementation of the law.
Republicans argue that the law harms the economy, while Democrats point out that the law is already benefiting those with preexisting conditions who can’t be denied coverage, people younger than 26 who can stay on a parent’s insurance plan, and women who now pay less for preventive healthcare.
McConnell is up for a tough reelection race himself. His Tea Party primary opponent has criticized him for not supporting Sen. Ted CruzTed CruzLewandowski: Top Cruz aide advised Trump team before NH primary Five reasons why Donald Trump could be the 'Greatest Communicator' Victims of Nazi Art theft need Congress to HEAR MORE's (R-Texas) effort to defund ObamaCare by threatening a government shutdown.