Dominik Szoboszlai insists he has cleared the air with Liverpool team-mates Curtis Jones and Kostas Tsimikas, a week after a video appeared to show the trio squabbling over the captain's armband.
The row followed the 1-0 pre-season friendly win over Wrexham at the Yankee Stadium in New York.
Jones and Szoboszlai started arguing after the final whistle as the team walked across the pitch to greet the fans, and Tsimikas soon joined in.
Earlier in the match, Szoboszlai had handed the armband to Tsimikas rather than Jones when he was substituted. Cameras then caught the Greek left-back hurling it to the ground during the row.
Last Sunday's 4-2 defeat to Leeds told a different story, with Szoboszlai passing the armband to Jones.
The Hungary captain said: "It wasn't a discussion between the three of us. We just discussed something on the pitch. And then you (the media) blew it out of proportion. But next time we'll be able to sort it out in the dressing room. No one was angry with anyone else. Everything is clear, and things are fine."
With regular skipper Virgil van Dijk missing the United States tour and Joe Gomez injured in the opening match, Szoboszlai led the side for most of the trip.
New manager Andoni Iraola has yet to discuss the vice-captaincy, Szoboszlai revealed, a role left vacant by Andy Robertson's move to Tottenham Hotspur.
He added, as quoted by the BBC: "No, there hasn't been any discussion. Obviously we have Virgil as captain. We have Alisson. We have Joe Gomez. So there hasn't been any discussion at all. Obviously Alisson isn't here and Joe is injured. It was a moment of great pride, and let's see. It's not in my hands. It hasn't been decided yet."
The Hungarian continued: "If you were offered the role of vice-captain at Liverpool, you would accept it. It would be a moment of pride for me. But I respect everyone, so either way, whoever takes on this role, we'll follow him."
Szoboszlai told reporters in Chicago, shortly before the Reds flew home to the United Kingdom, that he "cannot wait" for the new season.
He went on: "I feel we are ready to move forward again, and we'll draw motivation from last season, because it wasn't the best, we didn't reach what we wanted, so we'll try again."
He concluded: "It's not my job to tell you who we should sign or whether we should sign any player. If the officials and the manager decide to sign a player, we'll be happy with that and we'll welcome him. But if they decide not to sign, then we'll have to do something special with this team."