Obama also leads in the so-called "superdelegate" race - senior Democratic party members who vote on the nomination at the party's August convention.
Voting ends in Kentucky at 7pm local time (2300 GMT) and Oregon's mail balloting will end at 8pm local time (0300 GMT).
Results are expected shortly afterwards.
Kentucky and Oregon hold a combined total of 103 delegates, according to the Associated Press news agency.
Obama currently holds 1,602 delegates and 299 superdelegates, while Clinton has 1,444 delegates and 280 superdelegates, NBC reports.
Al Jazeera's Mike Kirsch in Kentucky says that even if Obama clinches enough pledged delegates, he will not make any "victory" speech out of respect for Clinton and because there are still three more primaries to be held in the semi-autonomous territory of Puerto Rico, South Dakota and Montana.
|McCain has criticised Obama for "reckless|
judgment" on foreign policy [GALLO/GETTY]
Many in Obama's campaign are already looking ahead to the presidential election in November, where the Democratic victor will face John McCain, the presumptive Republican candidate.
"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.
Obama has targeted McCain during campaigning in recent weeks, condemning the Arizona senator for the influence of lobbyists in his campaign and defending McCain's criticism of his willingness to talk to leaders of hostile governments such as Iran without preconditions.
McCain on Monday condemned what he called Obama's "inexperience and
reckless judgment" after Obama said Iran did not pose the same global threat as the former Soviet Union had.
Obama, campaigning in the state of Montana on Monday, countered swiftly that if McCain was elected "we'll keep talking tough in Washington, while countries like Iran ignore our tough talk".
However, at a rally in Kentucky on Monday, Clinton told supporters that there was "no way" the race was going to end "any time soon".
The New York senator said that the party's superdelegates should reconsider her chances as she is the stronger candidate against McCain and has clinched primary victories in all the major US states, such as California, Pennsylvania and Ohio.
"There has been a lot of analysis about which of us is stronger to win against Senator McCain, and I believe I am the stronger candidate," she said.
Obama now holds his largest lead yet over Clinton, according to a Gallup poll released on Monday, with 55 per cent to her 39 per cent and a three per cent margin of error.
Clinton had held a 20 percentage point lead in the poll in mid-January.