"The Ways and Means Committee will be putting out a specific plan very shortly, and they're going to work with all of our members to look at and consider and address these concerns," Ryan told reporters Thursday. "I believe that the Ways and Means Committee will be working with these members in particular to find a solution."
Nearly all Republicans will get behind a bill to chop tax rates in an effort to spur economic growth. However, lawmakers have expressed concerns about other specific provisions in a tax framework released last month, most of which are intended to raise money.
GOP leaders have called for scrapping state and local tax deductions, a popular provision that largely benefits high-tax blue states. Some lawmakers in the Northeast opposed the budget resolution, but they never truly imperiled its passage.
Later Thursday, Ways and Means Committee Chairman Rep. Kevin Brady, R-Texas, told reporters that a meeting with the Republicans in favor of state and local deductions did not yield a compromise.
"They made it clear. They need this problem solved before they vote 'yes' on tax reform," he said.
Before the vote, the House Liberty Caucus, led by Rep. Justin Amash, R-Mich., also announced it would oppose what it called "wasteful" spending in the budget measure. The group raised concerns about deficits generated by it.
The budget also allows for tax cuts to add up to $1.5 trillion to the federal deficit over a decade.
Ryan on Thursday described the budget as "really important" to passing tax reform. It unlocks budget reconciliation, which allows the Senate to pass a plan with only a simple majority, or at least 50 of 52 GOP votes with Vice President Mike Pence breaking a tie.
Ryan did not answer questions about whether the House would make other controversial tweaks to the tax code, like changing the popular 401(k) contribution benefit. He deferred specific policy questions to the House Ways and Means Committee, the chamber's chief tax-writing panel.
Brady said the House will introduce a bill on Wednesday and start committee deliberations on Nov. 6.
— CNBC's Christina Wilkie contributed to this report.