Trump says Tom Homan, his former acting Immigration and Customs Enforcement director, will serve as "border tsar" in his incoming administration.
"I am pleased to announce that the Former ICE Director, and stalwart on Border Control, Tom Homan, will be joining the Trump Administration, in charge of our Nation's Borders," he wrote on his Truth Social site.
Homan was widely expected to rejoin Trump's second administration in a border-related role.
Trump said he had "no doubt" Homan "will do a fantastic, and long-awaited-for, job."
Meanwhile Republican candidates are vying for the role of Senate majority leader after winning control of the chamber.
Republicans are expected to hold at least 52 seats in the 100-member Senate after capturing three previously held by Democrats in West Virginia, Ohio and Montana in the November 5 election.
Current Senate Republican leader Mitch McConnell, who has led his party in the chamber since 2007, is not seeking election to the post.
"Rick Scott for Senate Majority Leader!" Musk, a tech billionaire who has emerged as a major backer of Trump in recent months, wrote in a social media post on Sunday.
Rick Scott for Senate Majority Leader! — Elon Musk (@elonmusk) https://t.co/lpT34yHTKkNovember 10, 2024
Trump has not yet endorsed anyone in the race for majority leader.
Musk is the world's wealthiest person. Scott, who represents Florida in the Senate, is a former healthcare executive and the wealthiest sitting senator.
In an interview on Fox News on Sunday, Scott said the Senate needs to implement real change.
"We can't keep doing what we're doing," Scott said. "That's what Donald Trump got elected to do, to be the change."
In a tweet on Sunday, Trump said anyone seeking to be Senate majority leader must agree not to stand in the way of temporary presidential appointments being made when the Senate is not in session.
During Senate sessions, senators exercising their constitutional power hold hearings and confirmation votes for presidential appointees such as cabinet and heads of massive government agencies including the Defence Department and Department of Health and Human Services.
Scott responded on social media platform X that he agreed with Trump's demand, which would enable the incoming president to quickly fill posts, even if only temporarily.
Any Republican Senator seeking the coveted LEADERSHIP position in the United States Senate must agree to Recess Appointments (in the Senate!), without which we will not be able to get people confirmed in a timely manner. Sometimes the votes can take two years, or more. This is…— Donald J. Trump (@realDonaldTrump) November 10, 2024
Two other Republican senators vying to lead the body - John Thune and John Cornyn - later posted their support for Trump's push to quickly fill appointments.
Trump campaigned on promises, among other things, to deport immigrants who are in the United States illegally, cut taxes, impose tariffs on international trading partners and loosen fiscal policy.
Senate Republican leadership elections are scheduled for Wednesday local time, shortly after the chamber returns to work following a recess for the election.
The outcome of two Senate races - in Pennsylvania and Arizona - has not yet been called by Edison Research.
Control of the US House of Representatives has not yet been decided. Republicans already have won 214 seats, according to Edison, just shy of the 218 needed to retain their current majority.
If Republicans lead both chambers, it would mean that most of Trump's agenda would have a much greater likelihood of winning congressional approval than if Democrats controlled one of them.
Partial results show Republicans have a narrow lead in nine of the remaining uncalled House races and Democrats in seven, according to a New York Times analysis, although many thousands of votes remain to be counted.
With AP