Bassey is coming in as a replacement for Manchester United-bound Lisandro Martinez.
Rangers are receiving an initial £19m transfer fee from Ajax and it could rise to £22m depending on add-ons, making it largest transfer fee they have accumulated for selling a player.
Rangers have made a huge profit on a player they bought for £230,000 from Leicester City two seasons ago.
Bassey made 65 appearances in total for Rangers, including 50 last season when Giovanni Van Bronchorst's side made it all the way to the Europa League final, losing to Eintracht Frankfurt, and won the Scottish Cup.
The Nigeria international was also part of the Rangers side which won the Scottish Premiership under Steven Gerrard in the 2020/2021 season.
"Rangers Football Club can confirm Calvin Bassey has joined Ajax in a deal which is the largest in Rangers' history," a club statement read.
"Calvin leaves with best wishes from everyone at Rangers and we thank him for his hard work and commitment over the past two seasons.
"The 22-year-old joined Rangers from Leicester City for a Training Compensation Fee in the summer of 2020, making 65 appearances for the Light Blues.
"Bassey played his part in Rangers' historic 55th league title victory, won the 2022 Scottish Cup, while also having a huge role in the incredible UEFA Europa League run to the final in Seville last season.
"We wish Calvin all the best for his future career."
Bassey is Ajax Amsterdam's third summer signing after the arrivals of Steven Bergwijn and Owen Wijndal.
Bassey has won six caps for the Nigeria national team.
