Currently, about $2 million of the Massachusetts group’s $21.5 million budget comes from reimbursements for services to Medicaid patients. That money covers treatments for some 10,000 patients a year, about a third of its total, Planned Parenthood officials said.

“At a time when extreme politicians in Congress want to block millions of people from accessing essential preventive care at Planned Parenthood health centers, it is reassuring to see Governor Baker put the health and well-being of our communities ahead of politics,” said Dr. Jennifer Childs-Roshak, chief executive of the Planned Parenthood League of Massachusetts.

As the governor of a liberal-leaning state that was the first in the nation to legalize gay marriage more than a decade ago, Mr. Baker has on social issues taken many positions to the left of national Republicans. Last year, he signed a law protecting the rights of transgender people, and he also supports abortion rights.

Even so, the willingness to step in to fill a potential budget gap for Planned Parenthood is a pronounced move for Mr. Baker, a former health care executive who has made reining in the state’s budget a major goal of his administration.