Kimmel opened up the discussion when he asked Sanders if he had seen Wednesday's show
, when the host told Trump he had a question from Sanders' campaign, asking if Trump would be willing to debate the Democratic hopeful.
"You saw the show last night? You saw what I did for you?" Kimmel asked Sanders.
"You made it possible for us to have a very interesting debate about two guys who look at the world very, very differently," Sanders replied.
Trump said Wednesday he wanted proceeds of such an event to go to charity.
"That could happen, I've been saying that should happen anyway," he added. "If he paid a nice sum for charity, I would love to do that."
Minutes later, Sanders responded on Twitter.
"Game on," Sanders wrote. "I look forward to debating Donald Trump in California before the June 7 primary."
The Sanders campaign -- which has been looking to debate Democratic front-runner Hillary Clinton before California's primary on June 7 -- continued to push the idea on Thursday, with Sanders himself broaching the topic at the top of remarks at a Revolt TV town hall on Thursday morning in Hollywood.
"Now I understand that yesterday on the Jimmy Kimmel show, which I'm going to be recording later today, Donald Trump has agreed to debate me. I look forward to that," he told an audience during the live-streamed event.
Later in the day, Sanders' campaign manager said he hopes Trump doesn't "chicken out" after the two talked about the possibility of a one-on-one debate, but the presumptive GOP front-runner said he'd "love to debate Bernie."
"We are ready to debate Donald Trump. We hope he will not chicken out. I think it will be great for America to see these two candidates and the different visions they have for America going forward," Sanders campaign manager Jeff Weaver told CNN's Wolf Blitzer. "What we'll have to see, Wolf, is does Donald Trump have the courage to get on the stage with Bernie Sanders. That remains to be seen."
Trump made it clear Thursday that he wants to follow through with the debate.
"I'd love to debate Bernie. He's a dream," the real estate magnate said in Bismarck, North Dakota. "If we can raise for maybe women's health issues or something. If we can raise $10 or $15 million for charity, which would be a very appropriate amount."
"I understand the television business very well. I think it would get high ratings," Trump added.
The Republican nominee said his team has been in conversations with several networks about hosting the debate.
"It should be in a big arena somewhere. And we can have a lot of fun with it. I'd love to debate Bernie" he said. "The problem with debating Bernie is he's going to lose. Because honestly his system is rigged. Just like our system is rigged."
Thursday night, Trump submitted a question for Kimmel to ask Sanders, questioning if the senator would run as an independent "when Debbie Wasserman-Schultz and the party bosses steal this nomination away from you?"
"Let me tell Mr. Trump: I really do appreciate his concern for me. I know that comes straight from his heart," Sanders said, laughing.
"But tell him that what I hope will happen is that, in fact, I will run against him as the Democratic nominee for president of the United States, and if I do, we're going to beat him and beat him bad. You can tell him that."
Sanders slams Clinton
During Kimmel's show Thursday night, Sanders watched a clip from Clinton's interview with CNN, where she declared she will be the "nominee for my party ... That is already done."
Sanders shot back at his Democratic rival.
"Just a tinge arrogance there I think," he said.
"Unfortunately, the people of Indiana a couple of weeks ago, the people of West Virginia, the people of Oregon, who gave us a pretty good victory, don't quite agree with Secretary Clinton's assessment and we split Kentucky," Sanders said.
"I kind of think that on June 7, people of California will have a message for Secretary Clinton."