The top Senate Republican said Monday that there should be no stigma to wearing face masks to slow the spread of the novel coronavirus as President Donald Trump continues to refuse to wear one in public.

The remarks by Senate Majority Leader Mitch McConnell, R-Ky., came one day after Vice President Mike Pence appeared at an event in Texas, one of the new coronavirus epicenters, where he urged Americans to don masks and wore one himself while not speaking.

"We must have no stigma - none - about wearing masks when we leave our homes and come near other people," McConnell said in Senate floor remarks Monday afternoon. "Wearing simple face coverings is not about protecting ourselves. It is about protecting everyone we encounter."

The virus has killed more than 123,000 people in the United States, and U.S. cases make up by far the largest share of the worldwide caseload. On Sunday, the U.S. surpassed 2.5 million confirmed cases amid a new spike in infections in Texas, Florida, California and other Southern and Western states.

McConnell contracted polio as a 2-year-old in Alabama, and he has spoken in recent months about how the coronavirus crisis has made him think about his own life as well as the fear Americans felt during the polio epidemic that hit the country decades ago.

McConnell has also stressed the importance of wearing face masks during appearances in Kentucky and in comments to reporters.

On his trip to Dallas on Sunday, Pence attended a briefing with Texas Republican Gov. Greg Abbott, White House coronavirus response coordinator Deborah Birx and Housing and Urban Development Secretary Ben Carson. All four were wearing face masks as they entered and left the meeting.

As the pandemic has swept across the U.S., members of the White House coronavirus task force have typically not worn masks and have stood close to each other at media briefings, and Trump has frequently ridiculed reporters and others for wearing face coverings.

The battle in Congress over face coverings has at times grown tense. Last week, House Majority Whip James Clyburn, D-S.C., who chairs the Select Subcommittee on the Coronavirus Crisis, threatened to end the practice of holding in-person hearings because some Republicans on the panel refused to wear masks.

"I will stay in the safety of my home as I would ask all of you to do," Clyburn said during the Friday hearing.

Republicans, several of whom had worn masks into the hearing room before taking them off, contended that they could practice social distancing safely while seated maskless at the dais.

On Monday, Clyburn sent a letter to the top Republican on the subcommittee, Rep. Steve Scalise of Louisiana, informing him that any member who does not wish to wear a mask may join the upcoming hearings via videoconference - but will not be allowed to participate in person.

In the letter, Cyburn noted that the Office of the Attending Physician has issued guidance stating that face coverings are required for House meetings held "in a limited enclosed space, such as a committee hearing room, for greater than 15 minutes."

"The Attending Physician's guidelines make clear that the mask requirement is in addition to - not an alternative to - social distancing guidelines," Clyburn said. He added: "Unfortunately, the Republican Members' refusal to wear masks undermined the safety of everyone in the hearing room."

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The Washington Post's Robert Costa and Annie Linskey contributed to this report.