A split decision success against Australian Daniel Geale at the weekend made Barnet fighter Barker Britain’s fourth world champion.
Murray has twice been on the wrong end of disputed decisions at world level, once when he drew with Felix Sturm in Germany and again in defeat to Sergio Martinez in April.
Mandatory challenger Sturm is the likely opponent when Barker makes his first title defence, possibly in December.
But Murray believes a future domestic contest could mark the start of a classic British era in a division also featuring the likes of Matthew Macklin and rising stars Billy Joe Saunders and John Ryder.
Murray said: “We’re both world-class fighters. He’s a world champion and I should have become one, arguably against Sturm and definitely against Martinez, so we’ve got the chance to get a good middleweight era going again.
“Me and Barker have both said we want to get involved in big British world title fights and him beating Geale makes that possible.
“We’ve got a chance here for us to put on one the biggest fights out there. It’d be a massive event and my Twitter has gone crazy with people asking about it since his win.
“It’s up to Eddie Hearn and Matchroom Sport whether it can happen, but I can’t see any reason why it shouldn’t.”
Murray watched on television as Barker claimed the title in Atlantic City.
He said: “Regardless of whether I get a shot or not, I’m made up for him. He fought his heart out against Geale and he achieved his goal and I couldn’t be more pleased.”