"After all the votes are counted, we're going to accept the outcome of the election," Cornyn told his supporters. "That's what we do in America. We need to accept the verdict of the voters."
"What we're doing now is trying to encourage people sitting on the sidelines who think, 'My vote doesn't count,' or, 'There's something I'd rather do.' There is nothing more important for the future of our state and country," he continued.
This is considered a departure from President Donald Trump, who has repeatedly refused to say whether he would agree to a peaceful transfer of power if he loses.
"I have to see. Look ... I have to see," Trump told moderator Chris Wallace during a wide-ranging interview in July. "No, I'm not going to just say yes. I'm not going to say no, and I didn't last time either."
The president threatened legal action to prevent ballots from being counted after Election Day in Pennsylvania Monday.
Trump and his reelection campaign are signaling they will pursue an aggressive legal strategy to try to prevent Pennsylvania from counting mailed ballots that are received in the three days after the election.
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The matter could find its way to the Supreme Court, especially if those ballots could tip the outcome in the battleground state.
The three-day extension was ordered by Pennsylvania's top court. The Supreme Court refused to block it, but several conservative justices have indicated they could revisit the issue after the election.
Pennsylvania Secretary of State Kathy Boockvar, a Democrat, already has told local elections officials to keep the late-arriving ballots separate, but also to count them. She acknowledged that a post-election court fight could change that.
Cornyn, a Republican, is facing challenger MJ Hegar for his U.S. Senate seat.
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"I'm wearing my butt-kicking boots today because we're going all over this state," Hegar told supporters.
Hegar, a Democrat, is an Air Force veteran.
She campaigned in San Antonio Monday.
The Associated Press contributed to this report.
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