Bellew shocked the heavyweight division earlier this month when he stepped up from cruiserweight to stop former world champion David Haye in the 11th round.
The WBC cruiserweight champion is expected to stay at heavyweight for the rest of his career, with promoter Eddie Hearn linking him with the likes of Deontay Wilder, Tyson Fury and Joseph Parker before he hangs up his gloves.
Bellew may be on the sidelines for the next five months due to a hand injury but Whyte has urged Hearn to make the fight as he looks to put his rival in his place.
He told Sky Sports: "I'm one of the highest ranked heavyweights across the board, outside the champions.
"Tony Bellew says 'I wouldn't fight you, because you picked David Haye to knock me out in six rounds'. I'm like, dude, even your sponsor on your shorts picked David Haye to knock you out. That's why you were 16-1 or something silly like that. Stop talking rubbish and let's fight.
"If you want to prove yourself as a true heavyweight then fight me.
"Me versus Bellew will make more money than any other fight he can get at the minute. Trust me, that's a fact. I'm a bigger name here than Joseph Parker or Deontay Wilder."