Mayor Eric Adams of New York City has consistently pushed a moderate brand of Democratic politics, prioritizing public safety, policing, pro-business policies and fiscal austerity.

Others in his party have pushed back — arguing that New York, where Democrats overwhelmingly outnumber Republicans, should be a model for progressive stances, setting the tone for other large cities.

The city’s $100 billion-plus budget will be the next major flashpoint. On Wednesday, the City Council’s powerful Progressive Caucus will release its list of spending priorities, laying down battle lines over which version of Democratic leadership prevails in New York.

The caucus will demand major investments in housing, schools and mental health services, and ask the mayor to rescind his recent order for most city agencies to cut spending by 4 percent.

The fight is similar to others being waged across the country, as cities weigh the best approach to address crime, homelessness and poverty. In Chicago, Brandon Johnson, the progressive Democratic candidate for mayor, won this month on a public safety message that went beyond policing, and focused on youth employment programs and mental health services.

Mr. Adams and his top political adviser, Evan Thies, each discounted the notion that the mayoral election in Chicago was a sign that Democratic values had shifted to the left, and that New York may follow suit.

When Lori Lightfoot, the outgoing mayor of Chicago, lost in the first round of voting last month, Mr. Adams said he did not view her loss as a warning sign for himself. “What happened in Chicago,” he said, “is a message for all over the country that people want to be safe.”

Mr. Thies sought to draw parallels between Mr. Adams, a former police officer in his second year in office, and Mayor-elect Johnson — even though Mr. Johnson’s opponent in the runoff, Paul Vallas, was more in the Adams mold of a Democratic law-and-order candidate.

“Both Mayor Adams and Mayor-elect Johnson ran campaigns that prioritized safety and justice, and were supported by large Black electorates,” Mr. Thies said in a statement.

Mr. Adams has called for changes to bail reform that would make it easier for judges to keep people in jail; brought back a contested plainclothes police unit that was disbanded under former mayor Bill de Blasio; removed homeless encampments from the streets; supported charter schools; resisted closing the Rikers Island jail complex by 2027; and questioned the separation of church and state — in an overwhelmingly Democratic city. He has also recently ordered the third round of agency budget cuts in 14 months, acknowledging that key city services could be affected.

Now the Progressive Caucus is responding in force, arguing that the mayor’s cuts are “draconian” and will hurt New Yorkers at a time when rents have soared and the city is increasingly unaffordable.

The caucus’s leaders have characterized the mayor’s budget cuts as a “manufactured crisis,” pointing to a financial forecast by the City Council that found that the city will have a tax revenue surplus of $2.4 billion this year compared to the projection from the mayor’s office. The caucus will press the mayor to restore funding for schools and libraries and to address a staffing crisis in city government that has contributed to delays in basic services like applying for food stamps.

Library leaders have called the mayor’s proposed budget cuts of $52 million “devastating,” suggesting that they could be forced to cut hours. Mr. Adams has also received criticism for halting the expansion of 3K for All, a popular preschool program for 3-year-olds and a signature policy of his predecessor, Mr. de Blasio. At the same time, some schools have faced significant budget cuts, which Mr. Adams blamed on declining enrollment.

“We are committed to fighting back against the mayor’s harmful, unnecessary budget cuts and will do everything in our power to meet the needs of working New Yorkers,” Lincoln Restler, a co-chair of the Progressive Caucus, said in an interview.

The 20-member caucus will call for $4 billion for affordable housing and $350 million on providing “right to counsel” services to tenants facing evictions; full funding for universal preschool for 3-year-olds; and no further budget cuts to individual schools. It will also push to open four new overdose prevention centers, bringing one to every borough, and for an increase in funding for mobile mental health treatment.

The caucus may itself be at an inflection point. Earlier this year, it lost 15 members in a battle over ideological purity, after its leaders created a new “statement of principles” calling for a reduction in “the size and scope” of the Police Department, prompting more moderate members to quit.

The new agenda from the progressive caucus does not mention funding for the Police Department — the issue that led to the member exodus in February.

The progressive movement has also faced headwinds nationally amid a rise in violent crime during the pandemic and a backlash to the “defund the police” movement. But recent mayoral victories by Mr. Johnson in Chicago and Karen Bass in Los Angeles against more conservative “tough on crime” candidates have created momentum for the progressive platform.

Mr. Adams has argued that he has the winning message for Democrats, and he is planning to assist President Biden’s re-election campaign as a member of a national advisory board. He has sent more police officers into the subway, allowing police overtime costs to soar to more than $472 million during this fiscal year. At the same time, he has also championed an affordable housing plan that would spend $22 billion over the next 10 years — a figure some housing experts and elected officials say is not nearly enough. He has also prioritized a summer youth jobs program and a plan to help New Yorkers with severe mental illness.

Still, after the city saw an influx of migrants from the southern border, Mr. Adams has ordered repeated budget cuts to offset the costs of the city’s migrant response and new labor contracts with city workers. The city is facing major budget deficits in the years ahead, including a $9.6 billion budget gap in 2025, according to the Citizens Budget Commission, an independent fiscal watchdog.

“We are making the right decisions — the smart decisions,” Mr. Adams said at a news conference this month.

The City Council speaker, Adrienne Adams, released her own list of priorities earlier this month, including $61.5 million to expand discount MetroCards to more New Yorkers. Ms. Adams is not as far to the left as the Progressive Caucus, and the members who are most closely aligned with her left the group this year.

But Ms. Adams has made clear that she opposes the mayor’s budget cuts, saying they risked “taking the city down a harmful, destabilizing path,” and she has pushed to adhere to the plan to close the Rikers Island jail complex by 2027.

Ms. Adams and the City Council must negotiate the budget with the mayor and reach a deal by June.

The Progressive Caucus was founded in 2010 to oppose the policies of Michael R. Bloomberg, the former mayor who also preached the gospel of fiscal belt tightening. Every member of the 51-member City Council is up for re-election this year, with the June 27 primary occurring around the time when a city budget deal is typically reached.

Shahana Hanif, a co-chair of the Progressive Caucus, said that members of her caucus had criticized the mayor’s recent announcements introducing new robot police dogs and a labor deal with police officers that could cost $5 billion. But the group decided to use its budget agenda to instead highlight proactive investments that will improve public safety, she said.

“Our focus right now is to expand the services that keep New Yorkers safe that are being drastically slashed by the mayor,” she said.