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Jones defended his 205-pound belt for the fourth time on Saturday, submitting Vitor Belfort in the fourth round of the UFC 152 main event.
Sonnen is expected to face Forrest Griffin at UFC 155 in December. Interest in a Jones-Silva fight has grown in the last month. Jones refused to accept a fight against Sonnen on eight-days notice at a since-canceled UFC 151 event on Sept. 1.
Following the UFC 152 press conference, White said he feels other matchups for Jones would work better than Sonnen, but he'll make the fight people want.
"I can tell you right now, I think they both want that fight," White said. "I just think there are other fights that make more sense.
"People do want to see it. If enough people want to see it, I guess I'll have to make it."
When questioned on whom he would like to face next, Jones told ESPN.com that Dan Henderson would be his first option. Henderson was Jones's originally scheduled opponent at UFC 151 but fell off the card due to injury.
"I feel like I have a little unfinished business to do with Dan Henderson," Jones said. "I would like to take on another great."
Jones is expected to visit the hospital early Sunday to have his right arm examined. The 25-year-old was told on Saturday he may have suffered nerve damage in his right bicep during an armbar attempt from Belfort during the fight.
In an interview on "SportsCenter" prior to UFC 152, Jones (17-1) said it would have been a bad decision to "give a guy the opportunity of a lifetime," referencing his decision to decline a fight with Sonnen.
Sonnen (27-12-1) has come up short in two bids for the middleweight title in the past two years. December will mark his first fight at 205 pounds since 2005.
White says he fully expects Sonnen's verbal shots at Jones to continue after the Griffin fight.
"He's going to call out Jones; there's no doubt about it," White said. "Everybody talks about him talking his way into fights, but you know that guy is showing up to fight and win.
"Chael comes to fight. He'll fight anybody. I know that about him and it's one of the things I respect about him."