WASHINGTON—Senate Majority Leader Mitch McConnell deepened his bid to persuade Roy Moore to drop out of the race for an open Alabama Senate seat after accusations the Republican candidate engaged in sexual misconduct with teenage girls.

At The Wall Street Journal CEO Council, the Kentucky Republican suggested he wanted Attorney General Jeff Sessions to run as a write-in candidate instead for his old seat, and he said that if Alabama voters elect Mr. Moore in the Dec. 12 vote, he would immediately face an ethics investigation....