Healey also reiterated her pleas to press the federal government for more aid, including a larger group shelter setting that would ease some of the burden on the shelter system, more equitable distribution of federal dollars among states and expedited work permits for migrants — particularly Haitians, who make up a large share of those in the shelter system.

“This again affirms my call to the Biden administration,” Healey said. “I think they know and understand clearly what it is that we are seeking and those discussions are continuing. I’m hopeful that they will result in action soon for our state. But in the meantime we can’t wait.”

Healey’s warning comes days after federal Homeland Security officials visited the state to assess the migrant situation and meet with senior members of the governor’s team, state and local officials and service providers.

The number of families in the state’s emergency shelter system has doubled in the past year to nearly 7,000 — or just over 23,000 individuals, about half of which the state estimates are migrants. And more are coming each day.

With traditional shelter settings long full, Healey declared a state of emergency in August and the state has for months been placing migrants and homeless families in hotels, motels and college dormitories in some 90 communities across the state.

“State and local budgets can only stretch so far,” Healey said. “It’s time for the federal government to step up and do the job that we need them to do.”

Healey said on Monday that the state does not have enough space, service providers or funds to “safely expand” the shelter system past 7,500 families, a number she expects to hit in just two weeks. After that, the state will effectively turn to a triage system in which families with high needs — such as health or safety risks — will be prioritized for placement. Others will be placed on a waiting list.

Asked specifically if her announcement is a signal to migrants not to come to Massachusetts, Healey said “it’s certainly a communication that we are reaching capacity and therefore don’t expect to be able to house people the way we’ve been able to house people in the existing infrastructure.”

The worsening migrant and shelter crisis has come to dominate the Massachusetts governor’s first nine months in office, straining her relationships with state and local officials and overshadowing her progress on her other priorities.

And frustrations with the Biden administration’s response — or lack thereof — to the deluge of migrants overwhelming Massachusetts and other Democratic strongholds have begun to outweigh the potential political consequences of publicly berating the president. Healey, a member of President Joe Biden’s national campaign advisory board, no longer resists calling out what she views as insufficient action from his administration. Massachusetts House Speaker, Ron Mariano, publicly ripped the president earlier this month, saying Biden “better start paying attention to this.”

But Massachusetts Democrats are also now turning their ire on a leaderless U.S. House, which remains without a speaker for a second week and is holding up $4 billion in supplemental funding Biden requested to help Homeland Security manage the southern border and expand lawful pathways to immigration.

“The cavalry is not on the other side of the hill,” Mariano, who met with DHS last week, told a local television station afterward.

And so Healey on Monday appointed a former Massachusetts National Guard leader, Lt. Gen. L. Scott Rice, to oversee the state’s emergency shelter system and related response. She also announced several steps her administration is taking to help provide pathways out of the shelter system to free up more space, including expanding rental assistance and launching two programs aimed at helping migrants get skills training while they wait for work authorization and then connecting them to jobs once they have it.

In the absence of more federal dollars, Healey requested $250 million from the state legislature last month to help prop up the shelter system. That’s on top of the $325 million the state already budgeted for the program through the end of the fiscal year next June. But lawmakers are sitting on her request as they struggle to understand the scope of the crisis.

The $250 million “will help, but it’s not going to solve our problem,” Mariano said in the WCVB interview. “It may not even get us to the end of this month. And we don’t see this stopping at the end of this month.”