A threat to derail a regionwide Metro funding plan dissolved in less than 24 hours as D.C. Council Chairman Phil Mendelson said Thursday that he wouldn’t let his objections to the deal prevent it from being approved.

Mendelson (D) and five other council members sent a letter to Mayor Muriel E. Bowser (D) late Wednesday saying the city should give Metro only $167 million a year in new, permanent funding. That was at odds with Bowser’s support of a Virginia plan, approved in a historic vote Saturday, calling for the District to contribute $178.5 million as its share of a total of $500 million.

Mendelson’s position triggered a quick backlash from Metro supporters, including business, labor and church groups, and he backed off Thursday morning.

“The District will support Metro funding,” Mendelson said in a tweet responding to a Washington Post story that disclosed the dispute. “I regret that a better deal for our taxpayers was not met. . . . There is nothing wrong with fighting for a better deal for taxpayers, but there will not be a fight in the Council.”

Two council members who signed the letter — Mary M. Cheh (D-Ward 3) and Brianne K. Nadeau (D-Ward 1) — said they also would support fulfilling the entire $178 million obligation.

Following the council’s shift, Bowser pledged in her State of the District speech Thursday evening that the District would go along with the Virginia plan.

“Along with the mayor of Metro, [board chairman] Jack Evans, we’re going all-in and fully funding Metro at $178 million — the amount the District needs to contribute to finally solve the dedicated revenue problem that has held the system back for years and years,” Bowser said.

With the District’s top leaders now backing the Virginia plan, attention turns to whether Maryland will go along. The Democratic-dominated House of Delegates has passed a bill that would give Metro $150 million a year, and Gov. Larry Hogan (R) has endorsed it. But the Virginia plan calls for Maryland to contribute $167 million.

Many Democratic legislators and Metro supporters in Annapolis are pushing for the Senate to raise Maryland’s contribution and then persuade the House and Hogan to support it.

The short-lived disagreement in the District sprang from the city’s longtime dissatisfaction with a Metro cost-sharing formula under which the city pays a somewhat larger share of the region’s subsidies than Maryland and Virginia.

The formula is based on factors that include population, ridership and number of Metro stations. It is a primary basis for the plan approved in Richmond, under which Virginia would give Metro $154 million a year on condition that Maryland provides $167 million and the District commits to $178 million.

The Mendelson letter reiterated previous arguments that the formula is unfair to the city, partly because the District has fewer residents than Northern Virginia and the Maryland suburbs and only about a third of Metro’s track miles and ridership.

The council members who signed the letter said the three jurisdictions should each contribute the same amount — $167 million annually — to reach the $500 million target Metro says is needed to ensure safety and reliability.

The letter also confirmed publicly that some of the new money would come from increases in the city’s sales and property taxes, and possibly other taxes. Details will be available no later than the release of Bowser’s proposed budget next week.

Virginia and Maryland have defended the cost-sharing formula, noting that the District has 40 Metro stations, compared with 26 in Maryland and 25 in Virginia. The formula will become more favorable for the District, and more costly for Virginia, when new stations open with completion of the second phase of the Silver Line, which will extend Metro to Dulles International Airport and into Loudoun County.

Bowser and council member Evans (D-Ward 2), who chairs both the council’s Finance and Revenue Committee and the Metro board, have supported the $178 million amount — although Bowser’s office did not confirm that publicly until late Wednesday.

They want to avoid scuttling a regional accord to provide Metro with dedicated funding for the first time since the subway opened in 1976. Virginia Gov. Ralph Northam (D) has signaled he will sign the Metro bill approved by the General Assembly, although he plans to propose amendments to change the sources of funding.

In addition to Mendelson, Cheh and Nadeau, the letter was signed by council members Kenyan R. McDuffie (D-Ward 5), Vincent C. Gray (D-Ward 7) and David Grosso (I-At Large).

Cheh, who chairs the council’s transportation committee, said she signed it in hope of encouraging Bowser to open negotiations with Hogan and Northam to get a better deal for the city. “The idea behind the letter was to give [Bowser] some basis to at least open the door to talking to the other jurisdictions,” Cheh said. “Now that the mayor has said we’re going to pay the higher amount, I’m not going to frustrate that. I do wish we had taken the opportunity to at least open the conversation about a fair allocation.”

Nadeau echoed that point, saying, “At the time I joined the letter, it seemed there was an opportunity to get a better deal for District taxpayers, but that window has closed.”

Grosso, Gray and McDuffie declined to comment.

Mendelson said in an interview, “Once the mayor said she was going with the higher number publicly, that’s the end of the debate.”

Mendelson provided more detail about his position in his tweets, saying that over 10 years, the District would pay $245 million more than Virginia, and $115 million more than Maryland.

“This is money that could go to critical D.C. services,” he tweeted.

Within hours, Mendelson’s letter had drawn strong criticism from Metro supporters who have been advocating more funding.

Clergy from the Washington Interfaith Network phoned and texted D.C. Council members to urge them to support full funding at $178 million.

Metro’s largest union, Amalgamated Transit Union Local 689, said Mendelson’s letter “will derail dedicated funding negotiations and is a disservice to riders and the taxpaying public who deserve a safe, affordable and reliable Metro.”

The union continued, “The conversation about the desperate need to secure dedicated funding has been ongoing for the last two years. Raising an issue on the funding formula after the passage of Virginia’s [Metro] funding bill is completely irresponsible and is an inexcusable failure to negotiate in good faith with regional partners.”

The Federal City Council, an influential business and civic group, faulted Mendelson for his position Wednesday and welcomed the shift Thursday.

“The [D.C. Council] Chairman’s willingness to support providing the District’s share, at the $178 million level, is a constructive step towards achieving true reform for Metro and sends a strong signal to prospective employers that D.C. is committed to its economic future,” said W. Edward Walter, the organization’s chairman.