“We cannot just walk away from our responsibilities,” New York Republican Peter King said on Tuesday
The US House is to hold a vote on a $60bn (£37bn) aid package for areas hit by “superstorm” Sandy.
House Speaker John Boehner agreed to a vote on the full package by 15 January after an outcry from lawmakers.
Representative Peter King said a decision not to hold a vote before the current Congress ended was “a knife in the back”, but tempered his criticism after a meeting with Mr Boehner.
The October storm killed at least 120 people and flooded East Coast areas.
The Senate passed the $60bn aid package last week, but legislation will need to be reintroduced once the new Congress convenes on Thursday.
Mr Boehner said in a statement that the House would vote on Friday on funding of $9bn to the National Flood Insurance Program, and vote on the rest of the funds on 15 January.
Governors speak outAngry Democrats and Republicans spoke out on Tuesday night, shortly after the lower chamber backed the fiscal cliff deal, when House leaders announced there would be no further votes before the new Congress was sworn in.
“For the speaker to just walk out is inexcusable,” Mr King told reporters. “It’s wrong and I’m saying that as a member of the Republican Party.”
New Jersey Governor Chris Christie also laid blame directly on Mr Boehner during a news conference on Wednesday, saying the speaker and House Republicans had shown “callous indifference” towards his state.
Mr Christie, a Republican, said he had been repeatedly assured that the lower chamber would hold a vote on the relief bill before Thursday.
“There is no reason for me at the moment to believe anything they tell me,” Mr Christie said.
Earlier released a joint statement with Democratic New York Governor Andrew Cuomo, saying there had been a “dereliction of duty” on the part of the House of Representatives.
In a statement, President Barack Obama urged Congress to pass the aid package on Wednesday.
“Our citizens are still trying to put their lives back together,” he said. “Our states are still trying to rebuild vital infrastructure.”
‘Reset button’Earlier, the House had considered a smaller aid package of $27bn for immediate relief needs with a possible $33bn amendment for longer-term projects.
Some House Republicans said the Senate bill contained spending on unrelated projects or infrastructure proposals that should be paid with other funds. But both the House and Senate bills will expire with the end of the 112th Congress.
VIDEO: Occupy Sandy’s social media mandate (Produced by the BBC’s David Botti)
“If we get into the next Congress, you have to hit the reset button,” Jon Runyan, a New Jersey Republican, told NBC, adding that the bill had been largely pushed aside by negotiations over avoiding the fiscal cliff.
“It passed the Senate in a bipartisan way,” Representative Nita Lowey, a New York Democrat, said. “And again, to me this is a real betrayal, a betrayal of the leadership of the Republican Party.”
More than $2bn has been spent in 11 states and the District of Columbia on Sandy relief. The Federal Emergency Management Agency (Fema) still has $4.3bn in a disaster relief fund, enough to pay for emergency recovery efforts into early spring.
According to the New Jersey Star-Ledger, Fema’s extended transitional housing assistance for Sandy victims will expire next week, leaving about 2,500 people without funds to pay for hotels or motels.
And residents of two of New York City’s hardest-hit neighbourhoods told WNYC that money to rebuild was imminently needed.
“The residents are counting on it,” Matthew Fleming told the broadcaster.