Fury completed a stunning seven-round demolition of knockout specialist Deontay Wilder in Las Vegas last month to collect the WBC title, while countryman Joshua holds the other three major heavyweight belts after avenging his defeat to Andy Ruiz with a comprehensive points win last December.
Usyk, who is like Fury undefeated and has a London 2012 gold medal in common with Joshua, cleaned out the cruiserweight division and has similar designs having stepped up to take on boxing's big men.
"I work hard on it," he told Sky Sports of his desire to hold all the heavyweight titles, with a scheduled May 23 bout against Dereck Chisora next on the agenda for the 33-year-old Ukrainian.
That fight, along with Joshua's IBF mandatory against Kubrat Pulev at Tottenham Hotspur Stadium on June 20 could yet fall victim to coronavirus cancellations.
Joshua's promoter Eddie Hearn this week said the Pulev bout being shelved could lead to an immediate showdown with Fury.
Usyk, who is next in line with the WBO to challenge Joshua, is not overly concerned with such machinations.
"I want to fight both," the quicksilver southpaw replied when asked if he had a preference out of the two reigning champions.
"The last thing I think about is who will fight who. No predictions."
— Hayemaker Boxing (@HayemakerBoxing) March 20, 2020
Usyk's step up to boxing's blue riband division has been checked by injury niggles, with a bicep problem delaying his heavyweight debut – a seventh-round stoppage of American journeyman Chazz Witherspoon that remains his only outing since knocking out Tony Bellew on a final cruiserweight assignment in November 2018.
"I need to stay active. I need to box," he added. "If your vocation is passive, it's not good. If you stay active, it's very good. This is my active vocation."
Photo: Richard Heathcote/Getty Images Caption: Oleksandr Usyk celebrates with the hardware after beating Tony Bellew.