Washington (CNN) -- Senate Democrats conceded Thursday they don't have the votes to pass the DREAM Act, a bill that would have offered a path to citizenship for some illegal immigrants who entered the United States as children.

Democrats voted to pull the measure from consideration, a move that jeopardizes the chances for passing the hotly contested bill during the current lame-duck session of Congress that ends in early January.

While supporters say the measure that passed the House on Wednesday could still come up, each passing day reduces the likelihood for introducing and debating the act as legislative leaders battle over priorities in the waning days of the session.

Senate Republicans opposed the bill, standing by their pledge to block any legislation during the lame-duck session until the chamber approves bills to extend the Bush tax cuts and fund the government.

The so-called Development, Relief and Education for Alien Minors Act would have affected immigrants who entered the United States illegally as children under the age of 16 and have lived in the country for at least five years. Other requirements include graduating from high-school or obtaining a General Education Development diploma and demonstrating "good moral character."

Even then, only a six-year conditional status would be awarded. Before moving to the next phase, the students would need to meet additional requirements -- attending college or serving in the military for at least two years, and passing criminal background checks.

Proponents, including President Barack Obama and Democratic leaders, say the bill offers legal standing to young people brought to the United States who have bettered themselves and served their new country, while opponents claim it is a form of amnesty for illegal immigrants.

A White House statement Thursday praised Senate Democratic leaders for pulling the bill so that the chamber can take up the version passed by the House.

Noting that eight House Republicans voted with Democrats to pass the measure, the statement by Press Secretary Robert Gibbs said the proposal "should get bipartisan support in the Senate as well, and in light of the vote in the House, this is the right way to move forward to get that."

Rep. Luis Gutierrez, D-Illinois, a major supporter of the DREAM Act, said Thursday that "we now have the weekend and into next week to launch a national mobilization to get the votes to enact this important bill."

CNN's Craig Broffmann and Ted Barrett contributed to this report.