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Two-time heavyweight champion Joshua ended a two-fight losing streak with last weekend's unanimous points triumph over Franklin at London's O2 Arena.
Joshua went eight months between losing his rematch with Oleksandr Usyk in Saudi Arabia and facing Franklin but he does not expect the gap to be as long this time around.
"It's been nine months since I was last in the ring and I don't want to leave it that long again," he told BBC's One Show.
"I want to get back in there in the next three months, I'd say, and just kind of get the ball rolling."
Tyson Fury has been touted as the most obvious next opponent for Joshua, who called out his countryman after overcoming Franklin 118-111, 117-111, 117-111.
"I know who the fans want – they said Fury. The ball is in his court," Joshua told DAZN in his in-ring interview.
"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."
Drawn-out talks between the camps of Joshua and Fury have broken down on multiple occasions in the past.
Joe Joyce and Dillian Whyte have both called out Joshua over the past week but the 33-year-old is still pondering his next opponent.
"I'm definitely going to be fighting in the next three months. But against who? It's still unknown," he said.