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.
Control of the House gives a party the power to initiate spending legislation and launch impeachment proceedings against officials.
Leading both chambers would mean that most of Trump’s agenda would have a greater likelihood of winning congressional approval than if Democrats controlled one of them.
Trump campaigned on promises, among other things, to “end inflation”, cut taxes and deport immigrants who are in the US illegally.
Meanwhile, rumours are swirling around who Trump will offer his top cabinet jobs to.
Homan, the former acting director of Immigration and Customs Enforcement (ICE), had been tipped to become the next homeland secretary.
In a post on his Truth Social platform, Trump said Homan would be “in charge of our Nation’s Borders (“The Border Czar”), including, but not limited to, the Southern Border, the Northern Border, all Maritime, and Aviation Security”.
He continued: “Likewise, Tom Homan will be in charge of all Deportation of Illegal Aliens back to their Country of Origin. Congratulations to Tom. I have no doubt he will do a fantastic, and long awaited for, job.”
In a statement to the New York Post, Trump also announced Stefanik as his pick for UN ambassador.
She confirmed her acceptance of the role to the same newspaper, saying she was “truly honoured”.
“During my conversation with President Trump, I shared how deeply humbled I am to accept his nomination and that I look forward to earning the support of my colleagues in the United States Senate,” she said.
Elsewhere, Susie Wiles has already been appointed as the 47th president’s chief of staff. In his election victory speech, Trump called her “the ice maiden” – a reference to her composure – and said she “likes to stay in the background”.
Other names in the running to join the administration are billionaire X owner Elon Musk, who played a key role in Trump’s campaign, and Robert F Kennedy Jr – who ran his own presidential campaign before endorsing Trump.
Over the weekend, Trump announced that Nikki Haley and Mike Pompeo – who both served in his first administration – would not be offered new positions when he returns to the White House in January.