Another Democrat in Kentucky is edging closer to a 2020 bid to challenge Senate Majority Leader Mitch McConnell.

State Rep. Charles Booker has launched an exploratory committee to get in the race with at least three other Democrats vying for a chance to topple the six-term Republican senator.

“It’s clear that Kentuckians are ready for a change and they’re ready for a movement,” Mr. Booker told The Louisville Courier-Journal. “My goal with this process is to make sure that we can build the infrastructure needed to catalyze that.”

Though a powerhouse in Washington, Mr. McConnell is considered immensely unpopular at home and potentially vulnerable in 2020. But he has overcome tough challenges before, including in 2014 when he ranked as one of the most vulnerable Senate incumbents on the ballot.

Mr. Booker, 35, is a first-term legislator who told the newspaper he will run on a far-left agenda that includes a “Green New Deal” environmentalist makeover of the economy, a “Medicare for All” government-run health care system, living wages and “taxing millionaires like Mitch McConnell.”

He would enter a Democratic race that has been dominated by Amy McGrath, a Marine fighter pilot who last quarter raised a record $10.7 million for her campaign.

Ms. McGrath ran unsuccessfully in 2018 to oust Rep. Andy Barr, Kentucky Republican. She was then recruited to take on Mr. McConnell by Senate Minority Leader Charles E. Schumer of New York and liberal activist groups.

Other Democratic candidates gong after Mr. McConnell include retired Marine Mike Broihier and pharmacy technician Steve Cox. Kentucky House Minority Leader Rocky Adkins and Kentucky Sports Radio host Matt Jones also are eyeing the race.

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