One of the biggest challenges involves a proposal to eliminate the federal deduction for state and local taxes (SALT), which analysts say would hit upper middle-class families in high income tax states such as New York, New Jersey and California. The states are home to enough House Republicans to stymie legislation.
Brady gave ground over the weekend, saying he would allow a deduction for some local taxes to remain.
"We are restoring an itemized property tax deduction to help taxpayers with local tax burdens," Brady said in another statement.
But the gesture appeared to do little to turn the tide of opposition to SALT's elimination.
"I'm not going to sign onto anything until the full package is fully analyzed by economists," Representative Peter King of New York told the Fox News program Sunday Morning Futures.
"The fact that we're getting it at the eleventh hour raises real issues with me," he added.
A lobby coalition representing state and local governments, realtors and public unions rejected Brady's statement outright, saying the move would "unfairly penalize taxpayers in states that rely significantly on income taxes."
House Republicans have also faced opposition from Trump and others after proposing to sharply curtail tax-free contributions to 401(k) programs and move retirement savings to a style of account that allows tax-free withdrawals, rather than the tax-exempt contributions that are popular with 401(k) investors.
House Republicans now say they could permit higher 401(k) contribution limits but continue to talk about tax-free withdrawals. "We will expand the amount that you can invest. But we'll also give you an option to actually not be taxed later in life," House Republican leader Kevin McCarthy told Fox News.
The current cap on annual 401(k) tax-free contributions is $18,000.