As the Senate majority leader, Mr McConnell plays a key role in determining whether or not Congress votes to appoint a successor to justice Ruth Bader Ginsburg before the 3 November presidential election.
He said in a statement on Friday that he intended to hold a vote on whoever president Donald Trump nominates to succeed Ginsburg, who died aged 87.
Speaking on Saturday, the president said a replacement would be made as soon as possible.
Those plans sparked calls of outrage over the weekend, with Democrats criticising Republican plans to make an appointment less than 50 days to the election.
According to local media reports, around 100 demonstrators gathered outside Mr McConnell’s home in Kentucky, holding signs such as “Ruth Sent Us,” and “No Ethics No Shame”.
Some chanted: “Hey-hey, ho-ho, Mitch McConnell has got to go,” reported WDRB-TV, as others shouted to “vote him out”.
It was unclear whether or not the Republican was at his home in Louisville, Kentucky, or in Washington on Saturday.
One protester was reported to have been arrested by Louisville Metro Police for disorderly conduct and improper parking, after she parked her vehicle outside a CVS pharmacy without entering as a customer.
Another person kicked a glass door and damaged it, the Courier Journal of Louisville reported, leading to the deployment of a special response team.
Police cleared the scene around two hours after demonstrators began occupying the space outside the senator’s home.