Jenks, entering his first season with Boston, has been in a back-and-forth exchange with his former White Sox manager since signing with the Red Sox as a free agent. Francona said on Sunday that he has reached out to Guillen and Chicago bench coach Joey Cora about the dispute.
"That's over," Francona said. "I spoke to Bobby, actually went back and forth with Ozzie, not speaking, but messages. I'm confident that will be over.
"I don't care if they like each other, but Bobby is a Red Sox and we need to move on. I think [Guillen] understands that.
"Just making sure it goes away. It doesn't need to happen."
Responding to comments by Jenks in the Chicago Tribune that were critical of Guillen and the White Sox front office, Guillen said Saturday, "I wish I was mad about it because I will rip his throat [out]. That's sad because coming from him that surprises me. Everybody in this organization do a lot of great things for him.
"Too bad that all the stuff we had between me and Kenny interrupted his career because he did a lot of bad things last year," Guillen added, referring to White Sox general manager Ken Williams. "We lied for him, we protected him. I'm the first manager in the history of baseball to give a guy a week off to take care of his kids when his father-in-law was sick. … I don't think any manager is doing that.
"I bet you Tito Francona won't put up with the [expletive] we put up with here," Guillen said. "We don't miss him. You ask 30 guys in there."
In December, Jenks said it was going to be nice to go to Boston and play for a manager who knows how to run a bullpen. That led to a response on Twitter by Guillen's son Oney.
After Sunday, Jenks knows that his manager in Boston has his back. The pitcher addressed the situation one last time.
"It is what it is," Jenks said Sunday. "At this point I'm tired of it and I want to move on. I'm a Red Sox now and that's all I'm going to worry about. From this point on, I've got nothing else to say. He's going to have his say -- whatever. I'm a Red Sox and that's all that matters."
Jenks added that he wants to concentrate on baseball.
"It's very important, obviously," he said. "I need to get ready for the season and get ready for a championship year. That's why we're all here in this clubhouse and to put all my focus into that is going to be hard enough as it is.
"Hopefully this is the end of it. I'm sure later on in the season, once we get closer to seeing Chicago, I'm sure questions will come up again. Even then, I'm going to let it be."
The Red Sox host the White Sox in a three-game set beginning May 30 at Fenway Park.
Although Francona contacted Guillen, Jenks said he has no plans on reaching out to his former manager.
"It's done," Jenks said. "If he has anything else to say, that's up to him. On my part, I'm just going to walk away from this."
Of his relationship with Guillen and the White Sox organization, Jenks said, "It sucks, but it's a dirty game sometimes. It's part of the business, I guess. Moving on is the greater thing to do.
"Even before, I've always enjoyed playing for him. Like I said in the first interview, I'll always respect the man. He's a great person. My feelings still don't change on that matter. That's it.
"I loved Chicago. I lived there year-round and made it my home. That aside, my focus is on here now and to turn this year and next year into my main priority and do everything I can for this Red Sox team."
Joe McDonald covers the Red Sox for ESPNBoston.com.