The $1.9 billion Transbay Transit Center will open without its signature rooftop park unless the city of San Francisco can come up with $24 million in private donations and nonprofit grants to pay for the green space.

Cost overruns have added $300 million to the cost of the project at First and Mission streets, so money that was supposed to go toward building Rooftop Park - 4.5 acres with gardens, a children's play area and an amphitheater - will instead go toward constructing the four-story bus terminal and underground train station.

Transbay Joint Powers Authority spokesman Adam Alberti said the park would be built eventually, but that it will be "phased" and not necessarily ready when the transit center opens in late 2017.

"We know the park will come to fruition, just not with the public dollars," he said. "We are going to be relying more on private dollars."

The park's delay comes as the TJPA is eliminating a roster of design features to shave costs. The building's undulating glass skin was replaced with perforated aluminum to save about $17 million. An additional $53 million was scratched by eliminating the glass-fiber flecked finish from the walls. Fixtures that were to be stainless steel will instead be galvanized and painted steel.

On budget, on time

Alberti said all the "trade-offs" are aimed at "making sure the transit center comes in on budget and on time." He said the project's budget includes money for the park to be "roughed out" with drainage and walls. But there is no financing for the actual build-out - the trees, flowers, music venue and playground.

"We will aggressively look for park grants and other efforts to solicit funds from the public," he said.

With the dream of a bullet train from Los Angeles to San Francisco bogged down in a political, legal and financial morass, Rooftop Park is the most appealing amenity the project has going for it. It was designed by Peter Walker Partners Landscape Architects, and designer Adam Greenspan has described it as "interior spaces with rolling meadows and lawns with live oaks, redwoods, California buckeyes" and outer gardens that will be off limits to pedestrians so that "butterflies and birds and other kinds of visitors can have free rein of the scalloped alcoves without being disturbed by people walking through."

As planned, the park will be directly accessible from the fifth floor of neighboring towers and will feature a funicular to whisk visitors up from the street.

The park has been heavily promoted by developers constructing towers around the new terminal. The joint venture of Boston Properties and Hines, which is building the Salesforce Tower next to the terminal at 415 Mission St., touts the park as a major reason why tenants should locate in what will be the tallest tower west of Chicago. The same goes for developer Jay Paul, who is putting up an 802-foot residential and office tower that, like the Salesforce Tower, will connect directly to the park.

The park has its roots in a 2007 agreement between the TJPA and Hines, the Houston developer that teamed with architect Pelli Clarke Pelli and won a competition to design and build the transit center and adjacent tower. Hines initially offered $350 million for the right to develop the tower and agreed to build the park as well. But after the economic collapse of 2009, that number was reduced to $185 million, and the final agreement did not require Hines to build the park.

Park was a selling point

Former San Francisco Supervisor Chris Daly, a member of the TJPA board when the original agreement was struck, said the park was a crucial part of building support for the terminal and the neighboring private development around it.

"The park was always what we led with when talking about the project with the neighborhood," he said. "It was always envisioned as part of the capital project."

The uncertainty over the park comes as Transbay District developers are fighting with TJPA over how much they will have to pay as part of a Mello-Roos tax district designed to fund neighborhood infrastructure. Hines and Boston Properties, along with other property owners, are arguing that the district, as currently planned, is unfair. Hines and Boston Properties have also balked at being part of an additional community benefits district that would pay to maintain Rooftop Park once it's completed.

J.K. Dineen is a San Francisco Chronicle staff writer. E-mail: jkdineen@sfchronicle.com Twitter: @sfjkdineen