"It won't represent an expansion of the Hyde amendment," Cornyn said on the Senate floor. "It would basically maintain the status quo."
Sen. Dianne Feinstein was not immediately moved. "I want that language out," the California Democrat said of the Hyde Amendment after Cornyn unveiled his proposal. "There is a compromise possible, take it out."
Democratic Sen. Barbara Mikulski of Maryland appears to be on the same page. "I don't think that will work," she said of Cornyn's plan.
"I'm gonna look at it, but right now I'm skeptical," says Sen. Richard Blumenthal, D-Conn.
Republican Sen. Susan Collins of Maine took the floor after Cornyn and announced that she too would offer an alternative proposal after another vote on the Senate's trafficking bill.
Since last week, Democrats have argued that one of their issues with the Hyde Amendment language slipped into the trafficking bill is that it was added to an an authorization bill, which they claim is uncommon.
Cornyn's office argues that if the bill is defined as an appropriations bill, it might make it easier for Democrats to swallow the abortion rider, which has been attached to appropriations bills in some form since the 1970s.
(RELATED: McConnell Puts Democrats in a Box)
Democrats now have a hard choice to make. If they were to accept the offer, they could face a backlash from interest groups such as Planned Parenthood Action that have been calling for the rider to be removed.
But if they pass on Cornyn's alternative, it could continue to delay a vote on Loretta Lynch to be the next attorney general, a vote senators defending her say needs to happen sooner rather than later. Democrats only need four Republicans to cross the aisle and support her, and it appears Sens. Lindsey Graham, Orrin Hatch, Jeff Flake and Collins are prepared to do so. But if Democrats don't get a vote this week, the Senate could be wrapped up in the budget next week, pushing Lynch's nomination potentially until after the Easter recess.
"I want to get it done," Flake told National Journal of the Lynch nomination. "I just hope it moves forward."
Earlier Wednesday, Democrats were optimistic that they had the upper hand.
"Sooner or later Republicans are going to realize blocking her is a futile gesture that hurts them politically," Sen. Chuck Schumer said.
One Republican lawmaker worried that the GOP has picked the wrong fight.
"Eric Holder is ready to go. It is probably good for the country he move on. I think Republicans might be overplaying their hand here," the Republican Senator said on the condition of background so he could speak freely about leadership's strategy. "Most people see her as qualified. There is rightful anger on the president's executive order. I am not sure this is the right way to express it."