The D.C. Council gave its tentative but unanimous approval Tuesday to a bill that would funnel extra dollars to public schools serving low-income students and others at risk of academic failure.

It’s not yet clear how much additional money the city’s traditional and charter schools would receive — that’s up to Mayor Vincent C. Gray (D) to decide when budget season arrives next year. But there are signs that the investment, meant to help close persistent achievement gaps between poor and affluent children, could be large.

More than 40 percent of the city’s students fit the measure’s definition of “at risk,” and adequately supporting them could cost as much as $120 million a year, according to a study commissioned by Gray’s administration.

D.C. Council member David A. Catania (I-At Large), who introduced a version of the bill in June, said teachers and principals throughout the city have told him that they could do more for struggling students if they had more resources.

“They see firsthand what common sense tells us: Certain students require additional resources to support their education because of poverty and other barriers,” Catania said.

The council passed the bill without debate on first reading Tuesday and appears all but certain to give final approval next month. The measure has drawn broad support from parents, school leaders and education advocates, as well as from Gray, who called it a “good interim step forward” in a letter Tuesday to Council Chairman Phil Mendelson (D).

Gray and Catania are political rivals who have sparred over school policies since Catania took the helm of the council’s education committee in January. But the two said their staff members worked together to revise this bill, and Gray said the current version dovetails with his administration’s efforts to improve school funding.

Consultants working for Gray’s education deputy, Abigail Smith, have been studying the issue for a year and are expected to release final recommendations for changes this month.

The study’s draft recommendations, released in October, suggest that the city should invest an additional $88 million in helping at-risk students. But “at risk” was defined relatively narrowly and included fewer than 18,000 students, according to Catania.

His bill broadens the definition to capture more than 34,000 students. They include children who are homeless or in foster care or who qualify for food stamps or the welfare program known as Temporary Assistance for Needy Families. The category also includes high school students who are a year or more older than they should be for their grade level.

Council member Marion Barry (D-Ward 8) said Tuesday that he’d like to see an even broader definition. “If a person is left out that ought to be in, that means they don’t get the support,” Barry said.

If the bill becomes law, traditional and charter schools will receive additional funds based on the number of at-risk students they enroll.

The District has among the highest per-pupil spending rates in the country. Traditional and charter schools receive $9,306 for every student they enroll, plus extra money for students with disabilities and those learning English as a second language.

But activists have long argued that schools need additional dollars to address the needs of the city’s many poor children, who tend to lag behind their more affluent peers and often face profound challenges.

Other jurisdictions, including suburbs such as Fairfax County and cities such as Boston, Baltimore and New York, already provide extra resources for low-income students.

The bill requires the city’s traditional school system to send 90 percent of the additional at-risk funds to schools. Principals, in cooperation with parents, teachers and other community members, would then create a public document detailing how they plan to use the money.

Charter school leaders would have the latitude to use the extra funds however they want.

The bill also creates a grant program to spur stronger vocational education across the city, and it limits the amount of money that the central office can take from a school in any year to 5 percent of that school’s total budget.

More than two dozen parents and community activists signed a letter praising Catania and Smith for their work on the bill, which has changed significantly since it was introduced.

More funding for at-risk students “can prove to be an important step in strengthening educational opportunities for all of the children in our city, particularly those with the greatest need,” they wrote.

The council also gave its final approval Tuesday to two other education bills, one to end social promotion and the other to establish a new student advocate to help families navigate the city’s traditional and charter schools.

Both were among seven measures Catania introduced in June in what he described as an effort to accelerate school improvement.