Senate Republicans on Wednesday fast-tracked a House-passed bill that would restrict abortions after 20 weeks.

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"Senator [Mitch] McConnell's [R-Ky.] already talked about some pro-life legislation he'll be filing for cloture on at the end of next week," he told reporters.

That would mean that the Republican leader could tee up the abortion bill on Thursday, before the Senate leaves for the weekend, for a procedural vote early next week.

McConnell said earlier this year that the Senate would vote on the 20-week abortion bill, calling it "not only good news for pro-lifers, [but] good news for the entire country."

The bill could, however, face an uphill battle to overcome the 60 votes needed to overcome a procedural hurdle and split Senate Republicans.

But the vote could help take pressure off Republican leadership from conservative members who are demanding that a government spending bill cut off funding to Planned Parenthood, following a string of leaked videos showing the healthcare provider's staff discussing fetal tissue donation with undercover pro-life supporters.

Cornyn told reporters on Tuesday that the 20-week abortion ban bill is "an important vote for us to have, as we continue the investigations that Senator [Chuck] Grassley [R-Iowa] in the Judiciary Committee and others have into whether Planned Parenthood violated existing laws."

Lawmakers have until the end of the month to figure out how to fund the government and avoid a shutdown.

Democrats have slammed Senate Majority Leader McConnell for moving to the abortion bill instead of starting negotiations on how to fund the government.