Sen. Rand Paul Randal (Rand) Howard PaulSecond GOP senator to quarantine after exposure to coronavirus GOP senator to quarantine after coronavirus exposure The Hill's Morning Report - Sponsored by National Industries for the Blind - Trump seeks to flip 'Rage' narrative; Dems block COVID-19 bill MORE (R-Ky.) says Vice President Pence should oversee the Senate's debate on the GOP healthcare plan, pitching the idea as a solution to end the current stalemate in the House over replacing ObamaCare.

Paul suggested that a "firm commitment" that Pence would preside over the Senate — and be the one making decisions about what can be included in the bill — would help reassure conservative House members that their changes won't be stripped out by the Senate's parliamentarian.

Paul, speaking to reporters about what could help the House reach a deal, responded, "If President Trump and Vice President Pence made a public declaration that they are willing to make the vice president sit in the chair and have the vice president make the judgment on what is allowable under Senate rules."

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The move, which Republican Sen. Ted Cruz Rafael (Ted) Edward CruzLoeffler calls for hearing in wake of Netflix's 'Cuties' Health care in the crosshairs with new Trump Supreme Court list 'Parks and Rec' cast members hosting special reunion to raise money for Wisconsin Democrats MORE (Texas) has been pushing for weeks, would allow Pence to expand the scope of the legislation by effectively overruling the parliamentarian to bulk up the legislation with measures supported by conservatives.

"Are we going to let the parliamentarian decide what should be in the bill or are we going to let the vice president?" Paul questioned during a call with reporters.

The vice president is also president of the Senate, though in modern times, the incumbent has rarely filled that role except to break tied votes in the chamber.

House Republicans postponed a vote Thursday on their bill, a major setback for Trump and Speaker Paul Ryan Paul Davis RyanAt indoor rally, Pence says election runs through Wisconsin Juan Williams: Breaking down the debates Peterson faces fight of his career in deep-red Minnesota district MORE (R-Wis.), who have been working to wrangle votes and exploring ways to secure conservative backing.

Conservatives argue that Pence can decide what is and what isn’t eligible under the special budget reconciliation process, which allows Republicans to pass legislation with a simple majority.

Asked if he and the Freedom Caucus had been making that demand during their meetings with the White House, Paul declined to speak for House lawmakers but said that "it's been part of the discussions when I've been in the room."

Paul signaled that he was concerned that if conservatives included a repeal of ObamaCare's insurance mandates in the House bill that the Senate parliamentarian will strip them from the bill because of the Byrd Rule.

To pass muster under the Byrd Rule, legislation using the reconciliation process must be primarily focused on addressing the deficit, which is why only provisions that have a budgetary impact may be included.

GOP senators said this week that they are already working with the parliamentarian to make sure their amendments comply with the rule and will be eligible to be included in the repeal and replace legislation.

Senate GOP leaders are also warning against expanding the House healthcare bill too broadly.

Asked about Pence or even the parliamentarian ignoring the Byrd Rule, Sen. John Cornyn John CornynAirline job cuts loom in battleground states Senate Republicans signal openness to working with Biden Hillicon Valley: DOJ indicts Chinese, Malaysian hackers accused of targeting over 100 organizations | GOP senators raise concerns over Oracle-TikTok deal | QAnon awareness jumps in new poll MORE (R-Texas) downplayed the chances of that happening.

"I'm deeply skeptical of that because the Byrd Rule is not just a precedent, it's not just a rule of the Senate, it's a law. ... What we need to do is spend our energy and time trying to build consensus around something that can pass," he said.

Asked if House conservatives should stop pitching changes that could go against the Senate's Byrd Rule, he replied, "The House doesn't take advice from the Senate."