Real Madrid is the best fit for Kylian Mbappe. Besides, Barcelona has players in that same position as Kylian Mbappe they won't want his services. Jaoa Felix is a good winger and fits up for that position quite well.
He should have joined last season but plans didn't go as plan, always, you'll have to plan for that plan that's not part of the plans.
Turning to Barcelona might be possible but still not in this case.
Gennaro Gattuso would be managing his first game in League One, after all round failures. I want to see him finally successfull with his job with Marseille. I'm hoping he gets better with his tactics.
Kylian Mbappe is an important player for PSG but they have no option than let him exits Parc des Princes stadium as a free agent. Real Madrid president, Florentino Pérez have always admire the Frenchman from day one when he was still in Monaco, but was yet to developed to the full-grown complete player this present day. Kylian Mbappe moving to La Liga would likely end up in Real Madrid because they're keen on signing him and scouting him on their rader for such long period of time. Barcelona have no interest in him, even though they did it would be a total struggle for them because they're having imbalance finances structure.
Mbappe is the most likely to join Real Madrid as a free agent. But I wouldn't be surprised if Mbappe renews his contract with PSG again. In addition, the two giants of the Premier League, Arsenal and Liverpool will definitely try to bring Mbappe to the team. They have financial capacity. Both these teams will surely offer Mbappe a smart salary to get him into the team. It remains to be seen whether Mbappe will opt for the Premier League or La Liga. Mbappe's arrival at Barcelona is highly unlikely. Mbappe will surely demand a big salary. Barcelona don't have the financial ability to pay any player a big salary right now. Financially, this club is very weak.