The Principality Stadium has hosted late-night boxing before, including Joe Calzaghe’s fight with Mikkel Kessler in November 2007 which began at 02:00.
Joshua has twice previously fought in the Welsh capital, defeating Carlos Takam in October 2017, as well as Joseph Parker a year later.
A spokesperson for the WRU, which owns and operates the Principality Stadium, said: “We’d be delighted to host this fight.
“We are very keen to attract more boxing to the stadium, which has a rich history in the sport, enhanced by its closeable roof, unrivalled sight lines and unique city centre location.”
November has been mooted as the preferred date, however, the fight would have to be scheduled around Wales international rugby matches against Japan, New Zealand and Australia in the Nations Championship.
Last month, Matchroom promoter Eddie Hearn, who represents Joshua, dismissed speculation the fight could be staged in the US.
He said the current agreement, signed by all parties, states the bout must take place in the United Kingdom.
Fury and Joshua are both set to return to the ring later this month before a potential meeting later in the year.
Former WBC heavyweight champion Fury, 37, will face Poland’s Mariusz Wach in Thailand on 24 July, while two-time world heavyweight champion Joshua, 36, takes on Kristian Prenga in Saudi Arabia on the following day.
