Whittingham, whose Utes host No. 5 Stanford Saturday at 4 p.m., said all three have the potential of playing because all three are producing yards.
Poole is averaging 5.2 yards a carry. Radley is averaging 6.5 yards, and York is averaging 3.8 yards.
"All three have different running styles, so we're trying to give all three opportunities," Whittingham said. "Until one takes the lead and separates himself, they'll all get carries."
While there is an adage that if a team doesn't have a clear starter then they don't have anyone, Whittingham denied that is the case with the running backs.
"I believe in that when it comes to quarterbacks," he said. "But we have a good group of running backs and they aren't playing poorly and their overall production is good. We'll see if they figure out how to separate themselves."
The Utes finished with a season-low 99 net rushing yards against the Bruins, with quarterback Travis Wilson losing three yards.
However, Whittingham wasn't concerned that Utah's rotating running backs hindered Utah's production. The three backs averaged 4.08 yards a carry.
"It's overblown or overrated when you talk about how many carries a guy gets," he said. "We have a saying around here: It's what you do with your carries, not how many carries you get.
"We have a good problem," Whittingham added. "It would be different if all three were averaging just three yards a carry. We didn't run it as effectively against UCLA as we would have liked, but we are still averaging almost 200 yards a game."
Of the three backs, Radley's emergence is the most surprising development. The 5-foot-8, 187-pound junior played mostly on special teams last year but earned the coaches' attention with a good spring.
"No one is going to outwork him, and he is finally getting an opportunity," Whittingham said. "He has great feet and vision and has a knack of knowing when to slide inside and when to bounce outside."
Even with a run game that struggled against the Bruins, the Utes are averaging 194.8 yards a game on the ground and 286.4 through the air.
Whittingham said he expects Utah to continue to be a balanced offense overall and perhaps in the run game as well.
"We don't have to settle on a back as long as we keep getting production out of them," he said. "There is no rush or concern to settle on one. We can keep using all of them and see what happens."
No. 5 Stanford at Utah
P Saturday, 4 p.m.
TV • Pac-12 Network