Nasukawa, 20, was beaten by former five-weight world champion Mayweather, 41, in only 140 seconds in an exhibition boxing match on 31 December, BBC Sport reported.
Mayweather beat McGregor last year in the Irishman's first boxing match.
McGregor, 30, said he wants to take on Nasukara "before this summer".
"I wish to go to Tokyo to face Tenshin Nasukawa in a Mixed Martial Arts exhibition bout," he wrote on Twitter.
"Please arrange this, this instant. Yours sincerely, the champ champ," he added in a tweet directed to the Ultimate Fighting Championship and his management company Paradigm Sports.
In a subsequent tweet he said, "I'm the real Django no holds barred."
McGregor's last UFC bout ended in defeat by Russia's Khabib Nurmagomedov in October.