"A new timetable for the flight -- now postponed until at least mid-2019 and likely longer -- hasn't been released" by the California-based company, said the report in the Wall Street Journal.
The reason for the delay is unclear, but is a "sign that technical and production challenges are disrupting founder Elon Musk's plans for human exploration of the solar system," said the report.
"SpaceX also is confronting industry doubts about market demand for its Falcon Heavy rocket," it said.
The tourists would ride aboard a Dragon capsule, hoisted into space on SpaceX's most powerful rocket to date, the Falcon Heavy, which made its first test flight only four months ago.
SpaceX spokesman James Gleeson was quoted as saying the company "is still planning to fly private individuals around the moon and there is growing interest from many customers."
Gleeson did not immediately respond to an AFP request for comment.
The United States has not sent astronauts to the Moon since NASA's Apollo missions of the 1960s and '70s.
The tourists, who have not been named, "have already paid a significant deposit," Musk added at the time.
SpaceX is also planning to send astronauts to the International Space Station later this year, marking the first time since the US space shuttle program ended in 2011 that people have blasted to space from US soil.
SpaceX has told NASA it is targeting the first astronaut test flight no earlier than December 2018, but some observers say the schedule could slip to 2019.
Election Results for Lok Sabha Election 2019 will be out on May 23. Get the latest election news and live updates on ndtv.com/elections. Catch all the action on NDTV Live. Like us on Facebook or follow us on Twitter and Instagram for news updates from each of the 543 parliamentary seats for the election 2019