Senators vying to become the next leader of the Republican conference pledged on Sunday to quickly push through President-elect Donald J. Trump’s appointees after he demanded on social media that they do so.

Senator Rick Scott of Florida was the first to make such a vow in an attempt to curry favor with Mr. Trump. Mr. Scott quickly picked up the endorsement of one of the president-elect’s biggest backers, the billionaire Elon Musk.

Not to be outdone, Senator John Thune of South Dakota, who is considered a front-runner in the race, released a statement saying that he, too, would push to swiftly staff Mr. Trump’s administration.

“One thing is clear: We must act quickly and decisively to get the president’s cabinet and other nominees in place as soon as possible to start delivering on the mandate we’ve been sent to execute, and all options are on the table to make that happen, including recess appointments,” Mr. Thune said.

Senator John Cornyn of Texas was not far behind.

“It is unacceptable for Senate Ds to blockade President @realDonaldTrump’s cabinet appointments,” he wrote on social media. “If they do, we will stay in session, including weekends, until they relent. Additionally, the Constitution expressly confers the power on the President to make recess appointments.”

The jockeying for the president-elect’s approval was the latest example of his unparalleled influence over Republican members of Congress.