Denmark international Fischer, 21, has been with the Amsterdam club since 2011 and scored 24 goals in 79 league games.
The forward visited Boro, who were promoted to the Premier League this season, for a medical and agreed terms.
"It's the greatest league in the world and I feel really ready," Fischer told Danish TV station TV2.
Eight-times capped Fischer has agreed a three-year deal.
Fischer emerged as a key player at Ajax under recently departed coach Frank De Boer, alongside compatriots Lasse Schone and Christian Eriksen - the latter whom he will play against in next season's top flight when Boro face Tottenham Hotspur.
Ajax winger Viktor Fischer will join Middlesbrough for around £3.8m at the start of the international transfer window next month, BBC Tees reports.