Erik ten Hag
Erik ten Hag exploded into the limelight after Ajax lit up the Champions League in 2018-19, defeating Real Madrid and Juventus on their way to the semi-final. Only an injury-time goal by Lucas Moura stopped them from reaching the final. The manner of their progress to the semi-final is what astounded the audience and won them admirers. They weren't smash-and-grab wins, but contests where they dominated and imposed their style upon their opponents, especially in Madrid and Turin.
Some of us know him from before that. He was the manager of FC Bayern II at the same time when Guardiola was at the helm. In Pep Confidencial, there are mentions of discussions between the two on tactics and player development at the club and in general. Having a keen football mind, it was natural that he was heavily influenced by Guardiola. When he moved to Ajax and implemented his philosophy, the results were for all to see.
He moved Ajax from the traditional 4-3-3 to a 4-2-3-1 using Frenkie de Jong's dribbling and passing prowess to devastating effect. While he may have moved from the traditional formation, he stayed true to the philosophy of positional play. Even after departures of Matthijs de Ligt and de Jong, Ajax have stayed competitive and continue to play swashbuckling football, seen in the 4-4 draw against Chelsea. With 9 men, they were looking to score the 5th goal rather than play for a draw at Stamford Bridge.
Barcelona have had an issue in away games in the Champions League ever since the days of Guardiola. Erik ten Hag may be an answer to these problems. Ajax haven't lost an away game in Europe since ten Hag was appointed.
While he has an enviable away record, Ajax have famously given up leads in Europe to Tottenham and more recently, Chelsea due to this brave approach. It is one of the problems that Barcelona is trying to solve. As a result, there are some question marks. Apart from that, the German media believes FC Bayern want him as manager of the first team. Since he has already worked at the club, it makes sense for him to prefer Bavaria.