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Joshua has not fought since failing to reclaim the WBA-Super heavyweight, IBF and WBO belts from Oleksandr Usyk last August, having first surrendered the titles to the Ukrainian in September 2021.
The 33-year-old, who has lost three of his last five fights after going unbeaten through his first 22 as a professional, has reportedly held talks with both Demsey McKean and Otto Wallin in recent weeks.
However, Franklin appears to have emerged as the favoured candidate for Joshua's next outing, which promoter Dmitriy Salita expects to take place at London's O2 Arena on 1 April.
"Jermaine is a world-class heavyweight from Michigan in America, who is on his way to bringing a big win back home," Salita said.
"We hope to get the deal over the finish line to help Jermaine bring his lifetime dream to fruition."
Franklin made his first appearance on British soil in November, earning plaudits for his performance in a majority decision loss to Dillian Whyte, the first defeat of his professional career.