The couple, both 36, exchanged vows in front of a "tiny group of loved ones," the outlet said.
The New York Post did not say when or where the nuptials took place, though it noted Swift's private jet recently spent time in Nashville, the city where she lived early in her career.
Swift's publicist did not immediately respond to a request for comment, and Reuters could not confirm the Page Six report.
As workers scrambled to prepare Madison Square Garden for what was expected to be a lavish celebration for the megastars, the pair showed some support for New York.
The singer's publicity team said Kelce and Swift had donated $US26 million to several charities in the city and elsewhere this week.
Neither has commented publicly on their plans, but several media outlets reported Swift and Kelce will hold a 100-person event at the sports arena on Friday AEST followed by a larger celebration in front of 1000 people on Saturday AEST.
A sign posted at an entrance to the venue on Friday advised that anyone entering from June 29 through July 3 for "an event" could have their photo taken. By entering, people were agreeing to maintain strict confidentiality, including not discussing its hosts or attendees, the sign said, according to an image captured by a Reuters journalist.
At Madison Square Garden, workers have been hauling in food and decor to make the whole place shimmer. News cameras captured foliage, a box marked "garden party" and another labelled lobster meat.
The venue's public calendar shows no events scheduled until Tuesday - a rare six-day stretch in a summer otherwise packed with concerts, with just the occasional night unbooked.
New York City already was abuzz with major happenings over the US Independence Day weekend. Tall ships will sail into New York Harbour to mark the 250th anniversary of the Declaration of Independence on Saturday, and a World Cup soccer match is set for Sunday in nearby New Jersey.
The big events coincide with high temperatures that prompted city officials to declare a heat emergency.
When asked about a potential Swift wedding, New York Mayor Zohran Mamdani used the moment to urge people to stay indoors to protect themselves from the extreme weather.
"If you happen to be getting married at MSG, you will be staying inside and staying cool, and I think it's a good example to set for the city at large," he told reporters earlier this week. Reuters confirmed that an event planning company had applied for a permit to close the streets around the arena from Thursday through midday Saturday. Fencing was being erected around the venue's entrances this week.
The venue sits above a major commuter hub, and some commuters and tourists said the preparations and street closures were a hassle for people trying to get around New York.
That was not the case for Swift fan and Utah resident Tara Rosales, who said she felt "1000 per cent excitement" for the event.
The union of the Love Story singer and a Kansas City Chiefs star, dubbed "America's royal wedding," is one of the most anticipated celebrity weddings of the century.
Speculation has intensified since the couple announced their engagement in August following a public courtship that enchanted fans. Cameras caught Swift cheering on Kelce at Chiefs games and followed him as he jetted around the world to her concerts.
A star-studded crowd is expected, given Swift's roster of celebrity friends, including Selena Gomez, Ed Sheeran, Emma Stone and Gigi Hadid. In one TV interview, Swift joked that she would invite "anyone that I've ever talked to."
