House Democrats who are preparing an overhaul of political and ethics laws, a top priority of the incoming majority, have acknowledged that a component aimed at restoring a key section of the Voting Rights Act may take longer than their speedy timeline for the bill.

Other pieces of the overhaul, which Democratic leaders have said they will designate as House bill 1 in the new Congress, could also run parallel to the main package as a way to garner bipartisan support in the Senate, said Rep. John Sarbanes, the Maryland Democrat who is crafting the bill.

House Democrats have said they want to move swiftly in January on the overhaul as a way to prove to voters that the party can tackle other matters, such as health care and gun control, without the perception that big donors have outsize influence.

But Sarbanes said the voting measure may require more time.

“We’re not going to put any fixed deadline on that,” Sarbanes said Thursday during a news call on the bill organized by the outside group End Citizens United.

He said the measure requires special handling. That’s because the proposal, modeled after an existing bill, aims to create a new formula to determine whether certain states or jurisdictions need to get pre-clearance from the Justice Department before revising their voting rules. The Supreme Court struck down the old pre-clearance formula in the 2013 Shelby County v. Holder case, calling it outdated.

“But I think it can be handled in a timely way, a deliberate way, in a period of weeks,” Sarbanes added.

Congress passed the original Voting Rights Act of 1965 to ensure that state and local governments did not enact laws that would disadvantage any class of voters based on their race.

The overhaul package already faces opposition in the Senate, with Majority Leader Mitch McConnell of Kentucky making clear that it’s “not going to go anywhere in the Senate,” as he told a Wall Street Journal CEO Forum last week.

Still, Sarbanes said House passage of the package could generate public pressure on senators. “Then Republicans are going to understand they should get with the program on this,” the Maryland lawmaker said.

He also said House Democrats may look at pulling out components of the package and running them “along side” it as a potential path to getting bipartisan support for at least some parts of the bill, sooner rather than later.

House Minority Leader Nancy Pelosi, the California Democrat most likely to be speaker next year, has pledged that the bill would be a top priority in the 116th Congress. Both she and the party’s No. 2 leader in the House, Maryland’s Steny H. Hoyer, say the incoming freshman Democrats have built momentum for the overhaul by championing the issues in their campaigns.

House Democratic leadership will work closely with Sarbanes, End Citizens United and others “to get this legislation passed in the opening weeks” of the new Congress, Hoyer said during the Thursday call.

Rep.-elect Colin Allred, a Texas Democrat who defeated Republican incumbent Pete Sessions in the 32nd District in the suburbs northeast of Dallas, said his work as a voting rights attorney, who also counseled clients on campaign finance laws, led him to run for office.

“This is a critical package for us,” Allred said.

Niels Lesniewski contributed to this report.