The Missouri Democrat is remembered for his opposition to the 2011 debt-ceiling deal, which he called a "sugar-coated Satan sandwich" on Twitter.
That deal included about $917 billion in spending cuts over 10 years and ordered a special Congressional supercommittee to produce a deficit-reduction plan.
The committee's failure is what will trigger sequestration, the across the board cuts set to hit Jan. 1, unless lawmakers act to stop it.
Republicans have long advocated for means testing within Medicare. Under the House GOP budget, wealthy taxpayers would receive means-tested subsidies to purchase private healthcare coverage.
For more on Cleaver's comments, click here.