Citing several sources, the newspaper said Kaine will take over the post this month and will lead the party at the same time as he serves the final year of his term as Virginia governor -- operating part-time from the state capital in Richmond, and then moving to full-time status in 2010.
State law prohibits Kaine from serving a second consecutive term as governor.
Kaine, who was said to be under consideration as President-elect Barack Obama's running mate in the 2008 presidential election, is widely credited with helping Obama win Virginia, putting the state in the Democratic Party's win column for the first time since 1964.
Officials with the Obama transition team declined to comment on the report, and Kaine aides did not respond to requests for comment, the Post said.
The newspaper's sources said Obama will announce this week that Kaine will become chairman of the DNC.
After Obama won the November election, Kaine said he would not serve as DNC chairman, the Post said.
"I don't view that, frankly, as consistent with being governor, so I'm going to be governor," Kaine told reporters at a Richmond news conference. "I would view it as taking my eye too much off the ball about things that need to happen here."
Sources told the Post Obama pressured Kaine in December to take the job.