* Senate Democrats give strong support for deal
* Congress will miss own "fiscal cliff" deadline
* Agreement raises taxes on household incomes over $450,000
* House likely to be more wary of accord
WASHINGTON, Dec 31 (Reuters) - The White House and Senate leaders struck a bipartisan deal on Monday to try to avoid a "fiscal cliff" budget crisis, although the agreement was likely to face stiff challenges in the House of Representatives.
Senators were due to vote on the accord overnight and independent Senator Joe Lieberman said it had strong support from the Democrats who control the chamber.
The agreement came too late for Congress to meet its own deadline of New Year's Eve to pass laws halting $600 billion in tax hikes and spending cuts due to come into force on Jan. 1.
But with Tuesday a holiday, Congress still had time to draw up legislation, approve it and backdate it to avoid the harsh fiscal measures coming into force.
That will need the backing of the House where many of the Republicans who control the chamber complain that President Barack Obama has shown little interest in cutting government spending to try to reduce the U.S. budget deficit.
House Republicans are also likely to balk at planned tax hikes on household incomes over $450,000 a year that was part of the agreement struck between Vice President Joe Biden and Senate Republican Minority Leader Mitch McConnell. The House has convened a session for Tuesday at noon (1700 GMT).
House Speaker John Boehner said the House would consider the deal if it were passed by the Senate.
"The House will honor its commitment to consider the Senate agreement if it is passed. Decisions about whether the House will seek to accept or promptly amend the measure will not be made until House members ... have been able to review the legislation," Boehner and other House Republican leaders said in a statement.
The deal would make permanent the alternative minimum tax "patch" that was set to expire, protecting middle-income Americans from being taxed as if they were rich.
Indiscriminate spending cuts for defense and non-defense spending were simply postponed for two months.
As New Year's Day approached, members were thankful that financial markets were closed, giving them a second chance to return on Tuesday to try to blunt the worst effects of the fiscal mess.
There is no major difference whether a law is passed on Monday night, Tuesday or Wednesday. Legislation can be backdated to Jan. 1, for instance, said law firm K&L Gates partner Mary Burke Baker, who spent decades at the Internal Revenue Service.
"This is sort of like twins and one being born before midnight and one being born after. I think the date that matters is the day president signs the legislation," she said.
Republicans are pushing for savings in the Medicare and Social Security healthcare and retirement programs and threatening to block a increase in the debt limit - which caps how much debt the federal government can hold - in February unless they get their way.