The 34-year-old posted a photo of him and Mikitani in what appeared to be a private jet, along with emojis of the Japanese flag and a football.
Iniesta announced last month that he would be leaving Barca at the end of the season, bringing an end to a 22-year association with the club.
The midfielder has enjoyed a glittering career, winning eight La Liga and four Champions League titles.
Kobe currently sit sixth in the J-League first division standings after 15 games played, having finished ninth in the table last term.
If the move to Kobe is confirmed, Iniesta will join former Arsenal striker Lukas Podolski in the western Japanese port city after the German completed a move from Galatasaray last summer.
Danish legend Michael Laudrup, himself a former Barcelona fan favourite and one of Iniesta's boyhood heroes, also played for Vissel in 1996-97.
Iniesta, who famously scored the winning goal for Spain against the Netherlands in the 2010 World Cup final, was top of Mikitani's wishlist.
The billionaire businessman is also the founder and CEO of Rakuten, Barcelona's main sponsor.
Iniesta had also been linked to a move to Chinese Super League side Chongqing Lifan but they, like Vissel initially, denied the rumours.
If confirmed, the Spain legend's decision to choose Kobe would represent arguably Japanese football's biggest transfer coup with many big-name players now moving to big-spending Chinese clubs in the twilight of their careers.
It would also come as a timely boost to the J-League, which used to attract luminaries such as Brazilian great Zico and former England star Gary Lineker when it began in 1993 but has struggled to attract marquee players in recent years.
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