Ryan, interviewed by an ABC affiliate in Oxford, Ohio, on Thursday, said: "I'm joining the Romney ticket. It's not the other way around. So I'm supporting the Mitt Romney plan."
Ryan, asked if his budget plan could be merged with Romney's budget plan, replied: "Sure."
Clyburn, of South Carolina, pointed to Romney's public support of Ryan's budget in March, saying the former governor of Massachusetts has "doubled down in his support" by picking the seven-term congressman for his presidential ticket.
"I don't remember hearing Ryan say that these [budget cuts] will not be used," Clyburn said. "What I do remember is Romney saying that he believed in the Ryan plan. I remember that very clearly."
Ryan is due to stump Tuesday for votes in West Chester. Surrogates for the Obama campaign are scheduled to hold an event before Ryan arrives, again pitching the hypothetical education cuts.
The Obama campaign on Saturday circulated to reporters an email with links to 13 news reports, most of them citing Romney in March supporting Ryan's budget. The message: "Romney can't walk back his support for Romney-Ryan budget."
Romney unveiled his education plan, dubbed "A chance for every child," to the U.S. Chamber of Commerce in May.
That plan focuses on school vouchers, saying it takes "the unprecedented step of tying federal funds directly to dramatic reforms that expand parental choice, invest in innovation, and reward teachers for their results instead of their tenure."
The plan allows federal funds to be spent for tuition at public, charter, private or cyber schools.
"I don't see how you strengthen education by taking money out of the system and giving it to a few students to go to a private school," Clyburn said Monday.
Contact Chris Brennan at firstname.lastname@example.org or 215-854-5973. Follow him on Twitter @ChrisBrennanDN. Read his blog at phillyclout.com.