"We, as a club, don't play for the World Cup, we play for the MLS Cup," Rodriguez said.
Schweinsteiger then diplomatically responded, with a general reply about his competitive mindset.
"I always believed that you can win the game, even if it's against the best team in the world," he said. "Everything's possible."
The exchange immediately went viral, with the questioner being brutally mocked across social media as others cringed.
"Embarrassing ... Schweinsteiger was just asked if he can help the Chicago Fire win a World Cup," wrote Cristian Nyari on Twitter.
"I hope Schweinsteiger has a bonus clause in his contract for tolerating dumb questions," wrote another user.
When answering proper football questions, the German revealed his exit from Manchester United was so hasty, he never said a proper farewell to his colleagues.
"It was not so easy, because the decision was a bit late, so I couldn't say a proper goodbye to the team," Schweinsteiger said.
"Also they had a match away against - I can't remember - but they had a match on Sunday, so I couldn't really tell them, because I didn't want the focus on me, I wanted to have the focus on the team and the game. So I was very happy that they won.
"I had a great time there, especially with the team-mates and the staff there. It was a good time and I enjoyed it there."
Schweinsteiger said he is fit and ready to make his debut.
"I am ready," he added. "I was preparing before I came here and, of course, I need some days more training, but I feel I am ready, whenever the coach needs me.
"I need more training sessions to have the right rhythm, where I want to see myself, but my head is more than 100 per cent and I feel quite comfortable. The training session was pretty intense today and I could do it, so that was good."