Infographics on Cystic Fibrosis
Brown reportedly takes a personal interest in the issue because his youngest son Fraser, aged nearly two, has cystic fibrosis, which could one day benefit from embryo research.
But the Catholic Church and some opposition lawmakers are opposed to the bill, with one senior churchman warning it may lead to "Frankenstein" style experiments.
The Human Fertilisation and Embryology Bill would give scientists a legal framework to create hybrid embryos, yielding stem cells which could be used in research into treating conditions like Alzheimer's and Parkinson's.
Hybrid embryos are created by inserting the nuclei of a human cell into an animal egg and can ensure a more plentiful supply of stem cells.
The bill also backs the idea of "saviour sibling" children, who are created as a close genetic match for a chronically ill brother or sister, meaning their genetic material can help treat them.
In addition, it would give lesbians easier access to IVF treatment by removing requirements for clinics to consider a child's need for a father.
And it could tighten abortion laws by reducing the upper time limit for the procedure from 24 weeks to 22, 20 or 16 weeks if amendments tabled by backbench parliamentarians are supported.
Lawmakers have a free vote on sensitive parts of the bill, being debated and voted on in the House of Commons Monday and Tuesday, meaning their parties will not whip or take a line on it and they can follow their consciences.
Several Catholic Cabinet ministers are said to be among those agonising about whether to support it or not.
But Brown, who reportedly offered a free vote amid fears of a split, made clear his backing for the bill in an Observer newspaper article Sunday.
"I believe that we owe it to ourselves and future generations to introduce these measures and in particular to give our unequivocal backing, within the right framework of rules and standards, to stem cell research," he wrote.
"The question for me is not whether they (hybrid embryos) should exist, but how their use should be controlled."
The measures on hybrid embryos also have the backing of eight major British health charities whose heads signed an open letter to lawmakers published in Monday's Daily Telegraph newspaper.
"This approach could overcome the shortage of human eggs available for research," the signatories said.
"The overwhelming majority of those affected by conditions such as Parkinson's, Alzheimer's, motor neurone disease, multiple sclerosis, diabetes, cystic fibrosis and muscular dystrophy do not want any stone to be left unturned in the search for more effective treatments and cures."
The plan faces strong opposition, though -- in March, the Catholic leader in Scotland, Cardinal Keith O'Brien, called the production of animal-human hybrid embryos "hideous".
"It is difficult to imagine a single piece of legislation which more comprehensively attacks the sanctity and dignity of human life than this particular bill," he said in an Easter Sunday sermon.
"One might say that in our country, we are about to have a public government endorsement of experiments of Frankenstein proportion."
On Sunday, O'Brien and his counterparts in England and Wales plus Ireland announced a 25,000 pound (31,000 euro, 49,000 dollar) grant to help adult stem cell research.
They said too little time has been devoted to discussing this as an alternative to embryonic stem cell research.
When the bill passed its first parliamentary hurdle last week, only nine lawmakers from Brown's ruling Labour Party voted against it but the figure could be higher this time because that vote was whipped.
It comes during another difficult week for the beleaguered premier.
Opinion polls suggest his party faces a tough fight to avoid defeat in a key by-election Thursday in Crewe and Nantwich, north-west England.
The seat was a Labour stronghold until the death last month of veteran lawmaker Gwyneth Dunwoody but could fall to the main opposition Conservatives as Brown's support ratings slump.