The promoter Frank Maloney intends to approach Dunne's handlers with a view to setting up an all-British fight next year, once Schwer has recovered from Monday's Wembley points defeat by the World Boxing Council champion, Stevie Johnston.
Dunne, a former Newmarket stable lad, has not fought since defeating the South African Phillip Holiday last February and is overdue for a defence.
Schwer, from Luton, has a substantial following, as was evident at Wembley, while the Holloway-based Dunne can count on large support from Arsenal fans in north London.
The possibility of the fight taking place was discussed after Dunne landed the WBU crown two years ago, but that was before Schwer began to charge up the WBC rankings to the mandatory challenger's position.
"I'm going to speak to Dunne's promoter, John Hyland, about making that match," Maloney said. "Both boys are Brits and popular with the fans and their styles make for a classic confrontation.
"I thought Billy showed a performance of true courage against Johnston, but class came through. It was like meeting the great Roberto Duran in his prime. Had he fought any other lightweight champion than Johnston I think he would have won. Despite the margin of defeat, Billy has proved he is one of the best lightweights in the world. No one can argue that, forget the scorecards. Look what he was up against."
In the minutes after his fourth defeat in 42 fights Schwer's thoughts were geared to a new European title campaign. The championship is held by the Spaniard Oscar Garcia Cano, whom Schwer beat for the title in Zaragosa in 1997. If Maloney can reach agreement with Dunne's camp, Schwer might find that avenue more advantageous.
Schwer was cut over the left eye early on Monday, then on the right cheek, but the main problem was a jagged gash on the bridge of his nose, and his father, Billy, a veteran of 200 amateur bouts who was in his son's corner, said yesterday he wanted to pull him out of the title challenge three rounds early.
However, such decisions were the province of Schwer's manager, Mickey Duff, and his trainer, Jack Lindsay. And the referee Larry O'Connell did not give any indication that he would stop a fight which Johnston won by the landslide margin of 118-110 on all three judges' cards.
Schwer senior said: "I would have pulled him out in the ninth round but I never said anything. I could hear the punches hitting him. It was my son, and I could feel them the same as he was.
"I thought at that point he wasn't going to win - that was my opinion. I was proud of him in that last round."