A unanimous decision triumph against Franklin at the O2 Arena brought an end to a two-fight losing streak for Joshua, who swiftly set his sights on a bigger opportunity.
Joshua's clash against Franklin was his first non-title fight in 12 bouts and, if he gets his way, will return to familiar ground next time out.
Fury and Joshua have negotiated for a 'Battle of Britain' clash in previous years without securing an agreement but the 'Gypsy King's' failure to set up an undisputed bout with Oleksandr Usyk has opened the door again.
With Usyk expected to defend his belts against mandatory challenger Daniel Dubois, Fury's next move is not clear and Joshua has issued his challenge.
"I know who the fans want… They said Fury. The ball is in his court," Joshua told DAZN in the ring after his win.
"I would 100 per cent be honoured to compete for the WBC heavyweight championship of the world. I stand here and I say that proudly. It would be an honour.
"Wherever you are, if you're listening, you know my management, you know my promoter. We've had dialect before so let's continue and hopefully we can get this done sooner rather than later.
"We're not getting any younger.
"I can't wait to get back to Texas, to develop and push on. You know how one fight leads to another; I can't wait to get the next big fight."
Promising a new dawn against Franklin, Joshua fell short of expectations despite having the upper hand throughout, acknowledging he should have earned a stoppage.
"Someone else would knock him out, probably, but Jermaine has a good duck and dive style,"Joshus said.
"There were opportunities there but he knows how to tuck up. Respect to him, he done well, I respect him for that.
"Well done to him and his team for preparing. I should have knocked him out but what can I say now, it's done, onto the next."