Sen. Marco Rubio, R-Fla., plans to push more Republicans to support paid parental leave, a policy that has traditionally been a Democratic priority.

"We are hoping that we can continue to work to preach the gospel of paid family leave within our own party and sort of reframe the orthodoxy that has dominated this issue," Rubio said at a press conference Wednesday as he rolled out a retooled version of his parental leave bill, the New Parents Act.

More Republicans have joined Rubio in the mission already. The newest co-sponsor, who stood beside Rubio on Wednesday, was Sen. Mitt Romney, R-Utah. Reps. Ann Wagner, R-Mo., and Dan Crenshaw, R-Texas, were also present to introduce the House version of the bill. The legislation lets new parents receive Social Security early to pay for up to three months of leave in exchange for delaying retirement.

Rubio isn't the only Republican to back such a plan. The idea originally came from a white paper published by the conservative Independent Women's Forum, and has support from conservatives who want an optional paid leave program that wouldn't raise taxes on workers.

GOP Sens. Mike Lee of Utah and Joni Ernst of Iowa introduced a similar billearlier this month. Sen. Bill Cassidy, R-La., plans to unveil a third bill soon, which also uses Social Security. The lawmakers have met together and with White House adviser and first daughter Ivanka Trump. At least one other GOP senator, Todd Young of Indiana, attended a meeting on the issue.

"I think it's good that we now have at least three separate groups working on paid family leave within the Republican party," Rubio said. "This was an issue that five years ago was not discussed as part of Republican orthodoxy. I think it's an important development that we now have both parties talking and thinking about this issue."

Trump said in a statement to the Washington Examiner that it was "exciting to see momentum" on paid leave.

But Democrats have coalesced around an entirely different approach in the Family and Medical Insurance Leave Act, authored by Sen. Kirsten Gillibrand, D-N.Y., who is running for president. The legislation funds up to 12 weeks of paid family leave through a payroll tax, and applies not just to new parents but to people who need to care for a sick relative or need to take time off work to undergo treatment.

On Wednesday, Republicans said they would not support that approach. Wagner said she was speaking to two Democrats in the House, but wouldn't name them, though she said "one in particular is very interested." Rubio said he had tried to get Democrats on board, but that there was still work to be done among members of his own party.

"As Republicans we have an obligation to get our own house in order," Rubio said. "This is still not widely accepted in some of the circles that we come from because it's a new issue, so it will take some work to get more and more of our colleagues on board with the concept that the government should have any role to play in this regard."

He noted that in the past Republicans had supported an idea that would let people take out their Social Security funds to invest them in the stock market, and said they should therefore support the notion of investing in families, which he called "the most meaningful thing you could do for our country and its future."

The latest version of the New Parents Act has been changed slightly to have the option not just to delay retirement by between three to six months per child, but also alternatively to receive less in Social Security for the first five years of retirement, or to pay back the amount at a later time. The previous version, the Economic Security for New Parents Act, included only the first option.

Rubio said that the Ernst-Lee bill was "very good" but that there "were some differences." Mainly, the Rubio bill lets parents choose to go back to work part-time and allows them to transfer the leave time with their partners. Wagner said the intention was to give parents flexibility to decide what was best for them and to be able to work with their employers on a schedule.

"This is something entirely based upon the principle of personal responsibility, where people are able to bring forward Social Security benefits at a time when they really need them, when a young child is coming home from the hospital," Romney said.