The Gunners have been very connected to the much-sought-after Red Bull Salzburg midfielder.
But Sky Sports says the 20-year-old will join German club RB Leipzig in January.
He is expected to leave Salzburg after his departure from the Champions League in the stage.
The Austrian club finished third in Group A Atlético de Madrid and Bayern Munich.
Szoboszlai had piqued the interest of Bayern and Real Madrid.
Madrid technician Zinedine Zidane is said to have contacted him before this month.
The Bernabéu technician sought to point to the 20-year-old as a direct replacement for Isco.
Meanwhile, Arsenal planned to pay its clause in three separate installments.
The Gunners have been looking for a new midfielder since they missed the Houssem Aouar this summer.
Szoboszlai has 25 goals and 33 assists in matches for the Salzburg national team.
Last season he was named player of the season in the Austrian Bundesliga.
And he also began to have an effect for Hungary, scoring three times in his first 12 international matches.
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Red Bull Salzburg
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