With most of the votes counted in Kentucky, Hillary Clinton has been declared the unofficial winner by one state official.
In the night's other primary contest in Oregon, rival Bernie Sanders is ahead.
Front-runner Mrs Clinton is almost certain to secure the nomination in July, with a significant delegate lead.
Mrs Clinton has won 94% of delegates needed to win the nomination and has a total of 25 states to Mr Sanders' 19.
Alison Lundergan Grimes, chairwoman of the Kentucky State Board of Elections, told CNN that unofficial results confirmed that Clinton would narrowly win the state's primary contest.
The polls closed in Oregon at 2300 EST (0300 GMT) but early indications show Mr Sanders well ahead.
In the Republican race, Donald Trump won the party's only contest on Tuesday, in Oregon, which was no surprise as the last candidate in the race.
The Kentucky Democratic primary will award 60 delegates to go to the party's convention in Philadelphia while Oregon's primary will award 74.