Final votes on the budget are expected Wednesday afternoon.
A GOP measure is approved, but a separate amendment fails.
The two amendments passed by voice votes.
House Rules Committee Chairman Sessions claimed the law will cost $5 million per person.
Democrats have pledged to use their budget amendments to protect and bolster the program.
Senate Minority Leader Harry Reid (D-Nev.) suggested House legislation could get to the Senate late Thursday.
The president's budget will be offered as a substitute amendment Tuesday.
Senators voted 94-3 for an amendment from Sen. Bernie Sanders opposing Medicaid cuts.
Reading bills aloud on the House floor is typically a routine and monotonous process. But not so for Rep. Rob Woodall (R-Ga.).
The budget amendment to boost spending was rejected in a 45-52 vote.