Opponents have condemned it as unethical and immoral. Bush restricted funding for the research in August 2001.
Bush, whose support base includes conservative Christian voters who tend to oppose the use of stem cells taken from human embryos, used the only veto of his presidency to date to reject a bill stem cell research in July.
The White House reiterated Bush's intention to use his veto power, by stating that American taxpayers should not have to pay for research involving the "intentional destruction of human embryos."
'First 100 legislative hours'
The bill is part of a six-measure package that Democrats vowed to vote on during their "first 100 legislative hours" after winning control of congress from Bush's Republicans in November elections.
Already this week, the House passed two other bills in the Democrats' legislative package, one to bolster US security and the other to raise the federal minimum wage.
"This legislation does not seek to destroy life"
Steny Hoyer, House democratic leader
The stem cell bill now goes to the senate, where supporters believe it will pass with a two-thirds majority needed to override the veto.
The stem-cell debate often transcends party politics, with some anti-abortion Republicans strongly supporting the research. Thirty-seven Republicans backed the bill on Thursday, while 16 Democrats opposed it.
Sanctity of life
Steny Hoyer of Maryland, the house democratic leader, said: "I believe this legislation does not seek to destroy life."
"It seeks to preserve and protect life."
"We have a moral obligation to provide our scientific community with the tools it needs to save lives."
Many scientists view embryonic stem cells as the potential raw material for a new era of regenerative medicine, hoping to harness the unique qualities of the cells to repair damaged tissue.
Stem cells can be thought of as a 'master cell' for the body, capable of growing into various tissue and cell types. Those taken from days-old embryos are especially malleable but "adult" stem cells found in babies and adults have also shown promise.
Christopher Smith, a New Jersey Republican, favours research on stem cells not taken from embryos but opposes the current measure.
"Where will this all take us? If this bill were to be passed and signed into law, we would see the demise, the destruction, over time ... of millions of embryos."
The new bill would allow federal funding on research involving stem cell lines derived from embryos created at fertility clinics that would otherwise be thrown away because they are not needed to implant in a woman to make a baby.
Last year, the House passed the bill 235-193 before Bush's veto.