For Mr. Obama, painting the conservative lawmaker as a sort of wild-eyed wingman to Mr. Romney carries clear benefits, according to his advisers: it yokes Mr. Romney to the unpopular elements of the Ryan budget — from deep cuts in cherished social programs to a Medicare overhaul that could drive up costs for future retirees and fundamentally change the popular health plan — and it makes it tougher for Mr. Romney to tack to the center once he gets past the primaries.

“He’s very much lashed to Ryan and the House Republicans,” said David Axelrod, a top strategist for Mr. Obama. “They share an economic view and a view on the budget. By essentially embracing the framework of Ryan, Romney is also embracing the steps that would be necessary to implement it.”…

Advisers to Mr. Romney said they did not expect him to resist comparisons between his budget plan and Mr. Ryan’s. After all, they said, not only does Mr. Romney largely support the plan, but any discussion between Mr. Romney and the White House over the budget will play to Mr. Romney’s strength on the economy, they said…

Strategists inside and outside the campaign predicted that Mr. Obama would try to focus on line items in the Ryan budget, to argue that it would hurt women, children and the elderly. Mr. Romney’s challenge, they said, would be to keep the focus on the parts of the plan intended to spur economic growth.