ALBANY, N.Y. — Pressure is mounting on Gov. Kathy Hochul to alleviate the state’s politically volatile migrant crisis and the financial strain stretching New York City to the brink.

The answers could rest on Hochul’s state budget proposal Tuesday.

The high-stakes budget fight set to engulf Albany will also include debates over how to fund top-ticket items like health care and schools while eliminating a $4.3 billion budget gap.

Hochul, a moderate, will face a challenge from her left flank from progressive Democrats, who want a tax increase on rich New Yorkers. It’s a proposal she has already rejected.

Adding to the annual intrigue: The governor will unveil her budget plan the same day as Mayor Eric Adams will release his own fiscal proposal for New York City, an unusual move created in part by a quirk of the calendar.

Both officials, who have publicly worked well together, have intersecting fiscal problems.

The state faces multi-year budget gaps with expected spending running ahead of projected tax revenue. Adams has backed $3.7 billion in spending cuts impacting a variety of city services (he partially scaled back $37 million of those cuts this week).

Fiscal watchdogs warn not everyone will be happy when the dust settles.

“The mayor and the governor are both facing challenges with the demands for higher spending on so many different fronts,” Andrew Rein, the president of the Citizens Budget Commission, a think tank, said. “We can’t satisfy every single constituency on every single front.”

Hochul’s spending proposal will be the starting gun to a roughly three-month negotiation for how to spend more than $200 billion in taxpayer money.

The budget process in Albany is an intense, but often opaque period of horse trading typified by closed-door meetings between the state’s three most powerful elected leaders: The governor, the Assembly speaker and Senate majority leader.

The deadline for the budget to pass is April 1, the start of the state’s fiscal year. Budget negotiations can often go past that April 1 due date at the state Capitol. Last year, Hochul and her fellow Democrats in the Legislature did not finalize the budget until May 3, more than a month late.

But there may be more urgency to get a deal done with all 213 seats in the state Assembly and Senate up for election this year.

And how to address the migrant crisis looms. On Friday, Hochul said the city will get the support from the state that it needs to manage the influx.

The governor’s sole mention of the flow of migrants into New York during her State of the State address was in the form of a teaser for her budget announcement.

The issue, which voters in New York have cited as a key priority for state officials to address, didn’t receive a mention in a nearly 200-page policy book on her 2024 agenda. It was a surprising omission after months of focus by state and city officials over how to support migrant needs and their impact on government services.

Gliding over the issue, as well as concerns facing immigrant communities across the state, confounded advocates.

“The lack of mention of immigrants in the State of the State had us all taken aback,” Murad Awawdeh, the executive director of the New York Immigrant Coalition, said. “We want to make sure we end up in the budget. It was perplexing.”

Still, Adams, who has called the flow of more than 476,000 people since 2019 into New York City a budget buster given the enormous cost of housing and other services, displayed patience after watching Hochul’s address.

“We both agree this is a national problem and the national government must deal with this issue, but it’s in our lap right now,” Adams told reporters after Hochul’s address.

Hochul previously declared a state of emergency in response to the influx of migrants and has pushed the Biden administration for additional aid and resources. She is expected to meet with White House officials this month to discuss the issue.

State lawmakers and Hochul last year approved $1 billion to help the city with migrant support, funding that included housing, health care and legal services.

Adams has not revealed how much he wants from the state in the coming fiscal year to address the issue.

But he has met with the governor privately to discuss funding for migrants. He’s also publicly pushed for a “decompression” strategy meant to disburse migrants in other areas of the state — a plan that has been vehemently opposed by many local officials suburban and upstate communities.

Officials at the state and city levels have also pressed the federal government to address the issue, but a broad-based immigration agreement in the narrowly divided Congress is unlikely.

That has left states to address the issue.

Hochul’s top budget adviser has already indicated the budget will end funding for hotel stays for migrants, primarily in New York City. Instead, Hochul wants to concentrate spending on legal services and helping migrants find employment.

That could potentially be a boon for some upstate communities by addressing a long-time labor shortage.

“We have a lot of employers, we have over 460,000 job openings in New York and we have an influx of folks who want to work,” said Justin Wilcox, the executive director of the business-backed Upstate United.

Hochul has pledged to focus on New York’s nation-leading outmigration by addressing the cost of living in the state — a promise that she has linked to her opposition to raising taxes in the budget.

The no-new-tax stance has drawn fire from progressive Democrats in the state Legislature who want to raise personal income tax rates on wealthy people. New York last approved a tax rate increase in 2021, a move that affected people making more than $1 million and couples filing jointly who earn more than $2.1 million.

“It’s not right that we allow billionaires to escape with so much while the rest of New York has so little,” said Sen. Jabari Brisport, a Queens Democrat and one of the leading proponents in the Legislature for a tax increase.

Brisport and the progressive advocates who are pressing for the tax increase in the budget point to the governor’s quest for making the state more affordable, especially for middle-class people who struggle to pay rent or buy their first home.

“She campaigned heavily in Black communities, middle class Black communities, asking for their support,” Brisport said. “Those are the very people who can’t afford to stay here.”

But the affordability argument cuts both ways, Assembly Minority Leader Will Barclay, a Republican, said. He pointed to a bevy of taxes on health insurance and fees imposed by New York that also add to the cost of living.

Barclay is rooting for Hochul to avoid any tax hikes in a final budget deal.

“She’ll have my public support on that,” he said.

Key to the progressive argument this year to raise taxes on rich New Yorkers is what they believe is a the political upside for Hochul.

“There is a budget gap, but we can’t cut our way out of it and make New York affordable at the same time,” Alicé Nascimento, the policy director of the advocacy group New York Communities for Change, said. “To me, it’s good politics. I don’t understand how having an intense focus on crime and highlighting that as a central part of her platform makes people want to vote for Democrats.”

New York has benefited from federal pandemic aid that has since dried up. The last tax increase in 2021 generated billions of dollars in revenue that Hochul and state lawmakers agreed to pour into schools — satisfying a yearslong push by progressive education advocates.

But the state is facing a hangover from the spending. The gaps between what the state is taking in from taxes and its estimated spending will reach $9.5 billion next year and $7.7 billion the following year — which would be some of the biggest holes in a decade.

Still, Democratic lawmakers, who hold supermajorities in both chambers, do not want to slow spending.

“We’re in recovery from a once-in-a-century pandemic. We are fighting inflation all over,” said Sen. Zellnor Myrie, a Brooklyn lawmaker. “We have costs that are rising all over. This is not a time to claw back; this is a time to invest.”