The Barnet boxer claimed the IBF middleweight title on a points decision against Daniel Geale in Atlantic City last month and must now face Sturm before lining up any further bouts.
But already firmly on his radar is St Helens fighter Murray in a clash he feels can recreate a golden era of British boxing and help re-ignite interest in the sport.
"A lot of people don't understand that I have a mandatory defence first but I'd love a big domestic British fight against Murray," Barker told Sky Sports' website.
"I think Matthew Macklin is going to need to start rebuilding a little bit if he wants a shot but Murray is definitely in line.
"He was unlucky against (Sergio) Martinez so if he can get a fight in before, I'll be ready to face him in the summer.
"It's a fight that would capture the imagination of the British public and would rekindle the likes of the Benn-Eubank era.
"They were fantastic fighters and we're good fighters - it's what we want to do. It's not just about getting the British boxing fans involved, it's about getting the general public and sports fans involved and raising the profile of the sport and that's what I want to do."
In August Barker defied the odds after being knocked down by Geale in the sixth round to get back up and beat the Australian. It was his first world title.
But the 31-year-old is not content to stop there and he has his sights on more top-level success as he bids to make his mark on British boxing history - starting with a meeting with Sturm, most likely in December this year.
"First and foremost I have got the fight against Sturm," said Barker. "I am over the moon that I am champion of the world but that hunger is there. I want to go on and achieve more and I want to cement my place in history as having been a British boxing great - that means defending my title and that's what I plan to do.
"Sturm is a good fighter and that's the job at hand but I've not gone off the boil, I've kept myself ticking over and I have stepped up my training this week.
"I'm still waiting for all the details to be confirmed but it looks like Sturm in December. I'd like to fight in this country and it would be great for it to be a homecoming.
"A nice local event at Wembley Arena or Ally Pally or something like that would be lovely but I've got a fantastic team behind me in Tony Sims and Matchroom and they will do what's right for me but hopefully it will be a nice homecoming."