The National Employment Law Project in a statement Monday called on both congressional houses to consider and pass "the best possible version" of the bill as quickly as possible.
The Senate agreed Monday to move forward on the tax bill agreed to by President Obama and congressional Republicans.
Among other things, the bill would keep in place for two years all tax rates enacted during President George W. Bush's administration, extend unemployment insurance for 13 months, temporarily reduce the payroll tax by 2 percent, shield middle-class taxpayers from the alternative minimum tax and extend several tax credits included in the 2009 economic stimulus package.
"Although there are parts of this bill that will do little to stimulate growth but a lot to deepen debt, moving quickly to renew the (unemployment insurance) programs is critical to the economy and to the many families who would be eligible for this support," the group said.
Before the Senate began voting, White House press secretary Robert Gibbs said Obama was "encouraged by what he heard in Senate" in terms of acting on the bill.
"(There's) broad bipartisan support in the Senate; there's broad bipartisan support throughout the country," Gibbs said. "I think and believe it (the broad support) will be reflected in Senate vote."