Mr. Trump's campaign has poured cold water on this report.
“President Trump is the only one who decides who his running mate will be, and anyone who claims they know who he will choose is lying,” said Brian Hughes, a senior adviser to the Trump campaign.
Other candidates Trump may nominate for vice president include former Republican rivals Tim Scott and Doug Burgum.
Florida Sen. Marco Rubio is also expected to run, along with New York Rep. Elise Stefanik.
South Dakota Gov. Kristi Noem was seen as a front-runner, but she has come under fire in recent weeks for writing about shooting her dog.
Republicans could hold her accountable.
Mr. Biden has been running ads clearly trying to appeal to Ms. Haley's voters, who are disgusted with Mr. Trump.
“If you voted for Nikki Haley, Donald Trump doesn’t want your vote,” the ad claims between footage of Trump attacking Haley.
A potential upside for Ms. Haley is preserving her future plans to run for president. She has not supported Trump, and some Republicans could hold her accountable if he loses the election without her support.
Christopher Galdieri, a political science professor at St. Anselm's College, said a joint ticket would make sense if Trump were a “normal presidential candidate.”
Galdieri said Haley was too “independent” for a former president.
“I think he wants Todi.” [yes-man]. I don't think he wants anyone to disagree with him or argue with him,” he told the Telegraph.
President Trump tends to humiliate his opponents
But Trump's tendency to “humiliate” his opponents could make Haley an attractive candidate.
“He might consider it a big win to get her on the ticket given all the things she's said about him,” he added.
Trump has said he will choose a running mate near the Republican National Convention in July. The Telegraph has contacted Trump and Haley's campaigns for comment.