Democratic presidential candidate Bernie Sanders is hoping to keep his campaign alive with strong showings in the Kentucky and Oregon primaries.
Front-runner Hillary Clinton is almost certain to secure the nomination in July, with a significant delegate lead.
She has been campaigning in Kentucky, saying husband and former President Bill Clinton would take charge of revitalising the economy.
Both races could be fairly competitive, national polls predict.
Mrs Clinton has won 94% of delegates needed to win the nomination, a total of 24 states to Mr Sanders’ 19.
Republicans will vote in Oregon on Tuesday, but that race is all but decided, with front-runner Donald Trump having pushed out all of his competitors.
The Kentucky Democratic primary will award 60 delegates to go to the party’s convention in Philadelphia while Oregon’s primary will award 74.
Kentucky’s primary is closed, meaning only registered Democratic voters can participate.
In Oregon, voters cast ballots entirely by mail.
Pressure is rising on Mr Sanders, a senator from Vermont who has historically been an independent, not a Democrat, to drop out of the race.
Some Democrats worry that his presence is hurting their chances of beating Mr Trump, a billionaire businessman with no political experience, in the general election in the autumn.
Mr Sanders recently won primaries in Indiana and West Virginia, but that did not help him cut into Mrs Clinton’s delegate lead.
“I don’t think they think of the downside of this,” said Senator Dianne Feinstein, who supports Mrs Clinton.
“It’s actually harmful because she can’t make that general election pivot the way she should. Trump has made that pivot.”
Vice President Joe Biden has said he is confident Mrs Clinton will be the nominee.
Mr Sanders has argued that he still has a path to the Democratic nomination.
On the Republican side, Mr Trump is slowly gaining support among the GOP establishment.
He met with House Speaker Paul Ryan last week and the two had a “productive” conversation but Mr Ryan has yet to formally support him.
Mr Trump is only 103 delegates short of the 1,237 needed to clinch the Republican nomination and Mrs Clinton is 143 short of the 2,383 Democratic delegates she needs.