The Gypsy King has been ordered to defend his WBC heavyweight world title against the mandatory challenger Whyte, with the governing body ruling that Fury must take 80 per cent of the purse.
The rivals have a deadline of January 18 to come together for a private agreement for the fight and Arum believes that Whyte needs to agree a deal for the bout so a date and location can be arranged.
Arum, who is responsible for promoting Fury in the United States, told Sky Sports: "If he wants the fight and stops messing around, he can get a good payday and prove that he is of a world-class calibre by taking on Fury.
"We'd like to do that fight. It's a good fight for the UK.
"Whyte should sit down with myself and [Fury's UK promoter] Frank Warren and work out a deal for a guarantee, and we'll get this show on the road.
"Come to the table and let's see if we can carve out a real deal.
"Let's quit screwing around.
"I don't know Dillian but I would think he wants the fight. I don't know his psyche.
"Let's let Oleksandr Usyk fight Anthony Joshua, which is contracted, and let's get Fury to fight Whyte."
Fury has stated his intention to return to the ring in the UK on March 26 but Arum says his fighter will understand if it ends being slightly later.
He said: "If the fight is a week later or two weeks later? It's not the end of the world."