Two states -- California and New Jersey -- allow therapeutic cloning. California voters approved a plan last year to spend up to $3 billion over the next decade on the research.
The final version of the Massachusetts bill does not include any spending, although senators have suggested they might consider setting aside some tax dollars to encourage the research.
The bill was approved by a 119-38 vote in the House. The Senate approved it by a 34-2 vote last week. Both votes give supporters far more than the two-thirds majority needed to override a gubernatorial veto. A final procedural vote is needed before the bill is sent to Romney's desk.
The legislation -- a compromise between two versions approved last month in the House and Senate -- would allow scientists to create cloned embryos and extract stem cells for research.
Supporters say the research could lead to the cure or treatment of diseases such as diabetes and Parkinson's. But critics say it would allow scientists to create human life just to destroy it. They say other kinds of stem-cell research do not depend on the creation of embryos.
Romney supports research using adult stem cells or leftover frozen embryos from fertility clinics but opposes the creation of embryos.