Polls closed in Kentucky at 23:00 GMT on Tuesday while mail balloting in Oregon is set to end at 03:00 GMT on Wednesday with projected results from there expected shortly after.
Clinton's argument is based on her victories in Florida and Michigan.
The Democratic party stripped those states of their delegates because they violated party rules and held their primary polls early.
The Clinton camp has begun to include numbers from the two states in its calculation of how many delegates are needed to secure the nomination, saying that the total should be 2210.
But the party and Obama say 2026 delegates are needed.
The party is expected to decide the fate of Florida and Michigan's delegates on May 31.
Clinton says all 50 states should have a chance to cast ballots before the nominee is decided, with the last contest scheduled for June 3.
Clinton also claims she has the best chance of defeating McCain because she won in swing states such as Ohio, pivotal to a presidential election victory in November.
While Obama will still not have enough of the 2,026 nominating delegates overall needed to win the nomination after Tuesday's polls, he is likely to secure enough pledged delegates to make it almost impossible for Clinton to overcome his lead, analysts say.
Kentucky and Oregon hold a combined total of 103 delegates.
Obama also leads in the so-called "superdelegate" race - senior Democratic party members who vote on the nomination at the party's August convention.
But in a speech in Kentucky following her victory, Clinton vowed to battle on to end of the nominating contests on June 3.
"Some have said this was over, but not allowing everyone to vote would be a mistake," she said.
Obama, hoping to confirm his bid to be the party's candidate in November's presidential election showdown against John McCain, was set to hold his rally in Iowa, the scene of his shock win in the first nominating contest in January.
The Iowa event was to remind Americans "that this was a very unlikely journey that we've taken", Obama told MSNBC, while attacking McCain for offering a "third term" for George Bush, the current president.
Al Jazeera's Rob Reynolds said in Oregon that Obama was likely to win there as the state had a large middle-class and student population - groups that have supported the Illinois senator in the past.
Preliminary results from the Kentucky exit polls showed race remained an issue for many voters in the state.
Clinton won 70 per cent of white votes and three-quarters of white voters who have not finished college, The Associated Press reported.
In addition, only a third of Clinton's supporters said they would vote for Obama should he be nominated to face McCain in the presidential election.
Only 40 per cent of each candidate's backers want their candidate to pick their rival as a running mate for vice-president.
Olivia Ann Morris Fuchs, the director of Hillary Clinton's campaign in Kentucky denied Clinton's continued candidacy was hurting the Democrats.
"We're confident of [Clinton] being the nominee. She's ahead in the total popular vote and she has as many delegates as she needs," Fuchs told Al Jazeera.
"She's not hurting our party - we believe in every voice being heard and every vote counting."
Fuchs said Clinton would become the Democrat candidate for the presidency as only she could win in the key swing states like Pennsylvania and Ohio.
But many in Obama's campaign are already looking ahead to the presidential election in November.
"A clear majority of elected delegates will send an unmistakable message - the people have spoken and they are ready for change," David Plouffe, Obama's campaign manager said in a message to supporters.
The only primaries remaining are Puerto Rico, on June 1, followed two days later by South Dakota and Montana.
On Wednesday, both candidates are set to campaign in Florida ahead of the key Democratic party decision on its delegates on May 31.