The close call in the House Budget Committee for the Ryan budget led many to wonder whether the House leadership would have trouble with the bill on the floor. But The Hill reports that they’re making progress toward that goal:

Conservative House Republicans on Thursday said they support the 2013 budget resolution, leaving GOP leaders increasingly confident they will be able to pass the measure on the floor next week.

A day after the budget was approved in committee by a single vote (with two Republicans voting no), prominent members of the conservative Republican Study Committee said they back the resolution crafted by House Budget Committee Chairman Paul Ryan (R-Wis.).

RSC Chairman Jim Jordan (Ohio), who was undecided earlier this week, told The Hill he was won over by stronger limits on welfare and Medicaid spending.

RSC member Mick Mulvaney (S.C.), who also sits on the Budget Committee, said earlier this week that he might vote against the budget on the floor. But he ultimately backed the measure during the budget panel’s Wednesday markup, he said, after winning concessions from Ryan on increasing the panel’s oversight powers.

“Paul and I talked, and he agreed to make some additional accommodations to the RSC,” Mulvaney said.

Since the budget is a political document anyway, passage on the House floor is really the only hurdle. When two conservatives voted against the bill in committee and the Club for Growth opposed it, you could see trouble for passage, especially if Democrats kept their word and voted against it en masse. But Jordan’s support in particular probably brings along the overwhelming majority of hardliners, and I wouldn’t therefore expect more than token opposition next week.

The question then becomes why Republicans would want to force their most vulnerable members to walk the plank on something that has no chance of becoming law. The answer is that this is a rare circumstance where both parties think they have a story to tell. Democrats are obviously salivating at the prospect of a budget that they will say ends Medicare as we know it and eliminates a raft of cherished programs, with severe consequences for seniors and the poor. Democrats will also argue that Republicans broke the debt limit deal by forcing spending lower that the targets laid out there. Republicans want a budget to prove that they can do their job, and they want to accuse Democrats of ignoring the deficit and continuing down the path of big spending and big government. They also, they argue, “softened” the Medicare piece so that it’s more palatable to seniors, and they can say that they’re protecting Medicare before it goes broke.

The polling shows that the Ryan budget, properly explained, provokes a negative reaction. But few things are ever properly explained, and Republicans will probably have more money to do the explaining. So they’re not going to back up on this. Let’s see how it plays out.