(CNN) Members of Congress are pressing to change the rules to allow remote voting during the coronavirus pandemic, despite the resistance from their leaders, signaling that the outbreak sweeping the globe may not only require an unprecedented response from the Capitol but also change the essential nature of how it operates.

These Republicans and Democrats say that they should follow the federal government's guidance as the virus spreads to their own colleagues: In the past 24 hours, Reps. Mario Diaz-Balart and Ben McAdams became the first members of Congress to test positive for the coronavirus. Over two dozen members have already announced steps to self-quarantine or otherwise isolate themselves as a precaution.

Sens. Dick Durbin, an Illinois Democrat, and Rob Portman, an Ohio Republican, introduced a bipartisan resolution on Thursday to amend the rules to allow senators to vote remotely during a national crisis, giving the Senate's Republican and Democratic leaders the joint authority to allow it for up to 30 days. The Senate would then vote to renew remote voting every 30 days.

"We live in an age where national emergencies, public health crises and terrorism can threaten the ordinary course of Senate business," said Durbin. "We need to bring voting in the Senate into the 21st century so that our important work can continue even under extraordinary circumstances. Bob Dylan was right: 'The times they are a-changin'.' "

But Senate Majority Leader Mitch McConnell, a Kentucky Republican, has made clear both publicly and privately that he's against any such proposal, even though the Trump administration has advised limiting gatherings to 10 people. On Tuesday, McConnell said the chamber would "deal with the social distancing issue without fundamentally changing the Senate rules." He said the Senate could lengthen the time for a roll call vote to limit the number of people on the Senate floor at any one time.

Republican Sen. Roy Blunt of Missouri, the Senate Rules chairman, dismissed the growing calls on Thursday to change the rules, noting the objections of McConnell and House Speaker Nancy Pelosi. "It's not going to happen," Blunt told CNN. "We are going to continue to vote in person."

McConnell has pledged that the Senate will not leave town until it considers a trillion-dollar proposal from the administration to counter the coronavirus outbreak, larger than the federal government's stimulus package passed in response to the Great Recession. Still, some lawmakers forced to come to work say they are getting nervous about the growing threat. One senator asked simply, "Why are we still here?"

A bipartisan group of House lawmakers also requested a rule change that would allow for remote voting.

Rep. Katie Porter, a California Democrat, said she will send a second letter to House leadership on Friday with even stronger language calling for remote voting, arguing there is no legal, constitutional barrier to a rule change.

Reflecting increasing worry in light of the two House members being diagnosed with coronavirus, Porter told CNN in a phone interview Thursday evening that she will have more signatures than the 52 on the letter she sent earlier this week.

Like McConnell, Pelosi has pushed back on the idea. In a caucus meeting last week, the California Democrat rejected calls to leave town and vote remotely, according to a source familiar.

"We are the captains of the ship," she said. "We are the last to leave."

After passing two bills to address the crisis, the House went on recess. It plans to return when a deal is reached on the third, much more expensive measure. House Majority Leader Steny Hoyer told House Democrats on a conference call that he is willing to look at trying to pass the next stimulus package by unanimous consent, allowing members not to have to come back to vote, according to a member on the call. But the Maryland Democrat acknowledged, given the size and scope of the bill, it's "unlikely" that would be possible, according to the member.

Rep. Dan Kildee, a Democrat from Michigan, told CNN that the House leadership is having ongoing conversations about how to limit interactions on the floor, including extending the voting period.

"We have had some discussions about how we will manage the floor if we need to come back," said Kildee. "It is not just about the risk for ourselves, it is about setting an example for our staffs too."

On Thursday, Democratic Rep. Jim McGovern of Massachusetts, the House Rules chair, told his colleagues that he's doing a study on whether remote voting is feasible, according to a Democratic aide.

Rep. Tom Cole of Oklahoma, the ranking Republican member on the House Rules Committee, said he's skeptical of changing the rules. He argued that if negotiators achieve their goal of drafting a new coronavirus response package that can win a large consensus, remote voting won't be necessary, even with a number of members not voting because they are in quarantine.

"If some people can't come because they either have been exposed or in self quarantine, we still will have overwhelming numbers," Cole told CNN. "Nobody's going to throw rocks at you because you had coronavirus and couldn't come or you've been exposed to somebody and couldn't come. People will understand that, your constituents will."

Proponents of remote voting believe that their leaders will ultimately relent. A rules change would require the approval of two-thirds of senators but only a majority of House members.

If the efforts fail and Pelosi calls back members to vote in-person next week, Porter said, she'll assess the situation then and does not know now if she would show up to vote.

"I am going to do what is best for my health, the health of my family and the health of my community, including the folks who are working in our airports and work to serve the American people on my staff. If public health dictates that I stay away and I stay home, then that is what I will do."

Senators are bracing for one of their own to test positive for the novel coronavirus -- and if they do, that will almost certainly lead to the rule change, the proponents say.

"Once someone gets it, we're out of here," said a second senator granted anonymity.

This story has been updated with comments from Rep. Katie Porter on Thursday.