Andy Beshear, 41, the state attorney general and son of former Democratic Governor Steve Beshear, led by more than 10,000 votes with 98 percent of precincts reporting in a state that Trump won by 30 percentage points in 2016.
Bevin appeared poised on Tuesday to lose to Beshear, despite the state's strong Republican leanings and an election-eve rally headlined by U.S. President Donald Trump.
"With respect to our race, would it be a Bevin race if it wasn't a squeaker.
This is a close, close race.
We are not conceding this race by any stretch.
We want the process to be followed.
Know for a fact that there have been more than a few irregularities.
How many- which ones, affect will be determined according to law," said Bevin.
In a speech in Lexington, Kentucky, on Monday (November 4) night, Trump told voters that they needed to re-elect Bevin, or else pundits would say the president "suffered the greatest defeat in the history of the world."