Virginia Gov. Robert F. McDonnell (R) approved a deal Friday to help pay for the $2.8 billion second phase of the Dulles Metrorail line but stressed that he still expects the agency overseeing construction to meet certain conditions.

State officials, however, said they don’t intend to withhold Virginia’s contribution to the project.

In a Dec. 30 letter to the Metropolitan Washington Airports Authority (MWAA), McDonnell said he wants the authority to “fully comply” with a new federal law that gives Virginia more representation on the MWAA’s board. The board has opposed adding more seats.

McDonnell also asked for the “opportunity to examine all relevant proposed contract documents” on the second phase of the project to make sure they don’t violate Virginia’s right-to-work law.

In November, the airports authority approved a “project labor agreement.” That sort of deal usually dictates provisions such as work rules.

Some Virginia lawmakers introduced legislation in December to try to stop the state from contributing money to the project because they are concerned that the MWAA’s labor agreement would violate the state’s laws.

The first phase of the 23-mile rail line is under construction. It is expected to open in late 2013. Preliminary engineering on the second phase is scheduled to be done in February. A start date for construction of Phase 2 has not been set.

Under the terms for building the project, revenue comes from Dulles Toll Road users, Virginia, federal grants, the airports authority, and Loudoun and Fairfax counties.

Scott York, the Republican chairman of the Loudoun County Board of Supervisors, said the governor had “every right to force the issue” on the project’s labor agreement and getting more representation on the MWAA board. York said the MWAA board needs change, and the governor’s push is to “clean up the board, which has been inept, out of control and not listening very well.”

But Sharon Bulova, the Democratic chairman of the Fairfax County Board of Supervisors, said she hoped the governor “is not saying he’s withholding funding” pending the new seats on the MWAA board.

“Holding up the state’s share, which was there for a specific purpose to try to reduce the cost to toll road users, was critical in the memorandum of agreement,” she said.

“It is critical we move forward quickly. Time is money. The more we delay in allowing there to be agreement, the more it is going to cost the project, and the more it is going to cost the taxpayers,” Bulova added.

However, Martin Kent, McDonnell’s chief of staff, said the governor had signed and returned the memorandum of understanding Friday and does not plan to withhold funds.

“The commonwealth has agreed to pay the $150 million, but the governor raised several outstanding items related to the recently passed federal legislation that he asks be resolved as we move forward,’’ Kent said.

Transportation Secretary Ray LaHood declared victory in November after spending months negotiating a deal among the involved parties. As part of those negotiations, Virginia promised $150 million to help pay for phase 2 of the Dulles rail line project.

In his letter, McDonnell stated that the Dulles rail is a “very important project for the region, meaning that it is essential that, since the Commonwealth has not yet received written assurances concerning various legal and governance issues raised verbally and in recent correspondence, our performance under this Agreement remains subject to the following conditions.”

One of the conditions is that the MWAA “must fully comply in a timely manner” with recently passed legislation that increases the size of the 13-member board to 17, with Virginia gaining two seats, and Maryland and the District gaining one. Before the bill was passed, Virginia had five representatives, the District and the federal government had three, and Maryland had two. It also calls for the “automatic removal of all members upon expiration of their term.”

McDonnell wrote that he also wants the MWAA to “honor the decisions of each jurisdiction” in having the “authority under local, state and federal law to appoint members or remove their appointees for cause.”

The legislation was introduced earlier this year by Virginia Congressman Frank Wolf (R).

Sasha J. Johnson, a spokeswoman for the U.S. Department of Transportation, wrote Friday in an e-mail, “We are pleased Governor McDonnell, and all parties, have signed the memorandum of agreement which lays out everyone’s responsibilities.”

The incoming chairman of the MWAA board, Michael Curto, said he also looks forward to moving ahead.

“I am committed to working with each of the appointing authorities to secure the implementation of Congressman Wolf’s legislation,” he wrote in an e-mail, “and I am determined to see that the rail system progresses as quickly, safely and cost effectively as possible.”

McDonnell’s letter comes as users of the Dulles Toll Road will see toll road rate hikes. Starting Sunday, the toll rate on the Dulles Toll Road will increase by 25 cents to $1.50 at the main toll plaza.

The increase is the last of a series of annual rate hikes approved by the MWAA in 2009.

Revenue from the toll road goes to pay for operating and maintaining the highway and to helping pay for the new Metrorail line to Dulles.

Drivers along the Dulles Greenway, a private roadway that stretches into Loudoun County, will also see toll rate hikes. The maximum base toll on the Greenway will rise 30 cents to $4, and the congestion management toll imposed at rush hours also is increasing by 30 cents, to $4.80.

Watchdogs and some MWAA board members have expressed concern about the rising rates of tolls along the Dulles Toll Road.

They have worried that in 20 to 30 years, toll rates could rise to $17 round trip. Some MWAA board members have said the way to keep toll rates down is to get more money from other sources to help pay for the new rail line.