A party needs 218 seats to win a House majority. The Republicans have 215 compared to the rival Democrats' 210, according to CBS.
Control of the House gives a party the power to initiate spending legislation and launch impeachment proceedings against officials.
A majority in the House, the lower chamber of Congress, alongside the Senate, the upper chamber, would give Trump a greater chance of getting his plans approved than if Democrats controlled one or both.
In the meantime, attention has switched to who he appoints to serve in his administration.
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."
One of Trump's major campaign pledges was to deport immigrants who are in the US illegally.
A senior source close to Trump's transition told CBS News that Trump had picked Stefanik for ambassador to the UN, and Stefanik accepted the role. She confirmed her acceptance of the role to the New York Post, 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.
An official announcement has not yet been made but is expected soon.
Trump earlier appointed Susie Wiles as his chief of staff. In his election victory speech, Trump called her "the ice maiden" - a reference to her composure.
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.
There is speculation that Florida Senator Rick Scott could be in line to become Senate majority leader, having received the backing of Musk and others.
Trump has said Nikki Haley and Mike Pompeo - who both served in his first administration - will not be offered new positions.
Trump has confirmed that Homan will have a wide-ranging role overseeing border security and deportation policy - two of his main campaign issues.
Homan, a former police officer, also played an important role in Trump's first presidency as the acting director of Immigration and Customs Enforcement (Ice).
He resigned in 2018 but remained a proponent of Trump's hard-line approach. He supported separating migrant children from their parents as a means to deter illegal crossings, and has said politicians who support migrant sanctuary policies should be charged with crimes.
Homan is a visiting fellow at the Heritage Foundation - a conservative think-tank that published the Project 2025 document, which set out a "wish list" for a second Trump presidency. Trump has distanced himself from the organisation.
Stefanik became the youngest woman ever elected to Congress in 2014, then aged 30. She continues to represent New York’s 21st District.
After entering politics, the fourth-ranking House Republican initially positioned herself as a moderate and criticised Trump - but later became his loyal defender.
In his statement to the New York Post, Trump described her as "an incredibly strong, tough, and smart America First fighter".
The UN ambassador role is crucial to global cooperation and security, and Stefanik's nomination comes as conflicts continue to rage in Ukraine and the Middle East.
North America correspondent Anthony Zurcher makes sense of US politics in his twice weekly US Election Unspun newsletter. Readers in the UK can sign up here. Those outside the UK can sign up here.