Sen. Jim Inhofe, a Republican from Oklahoma, said he talked to Vice President Mike Pence about President Trump's criticisms of him, and Pence "was very upset with him."
Inhofe also called Trump's comments "regrettable."
Pence on Wednesday publicly broke with Trump, saying he cannot submit to demands he overthrow the results of the election.
Pence formalized his views in a letter to lawmakers, declaring he has no "unilateral authority to decide presidential contests" and could not change the results of the election.
"It is my considered judgement that my oath to support and defend the Constitution constrains me from claiming unilateral authority to determine which electoral votes should be counted and which should not," he wrote.
CNN's Kevin Liptak and Kaitlan Collins contributed to this report.