The Netherlands Under-19 international arrives at St James' Park after progressing through Ajax's youth system, having joined the Dutch club at the age of seven.
Ajax said in a statement that the transfer fee could amount to approximately €27 million ($31 million).
"It's an incredible feeling to be here. It's a giant club in the Premier League and it was always my dream to play in the best league in the world, so this is really exciting," Steur said.
“I am a player who likes to get on the ball and play forward all the time. I am happy playing between the lines and I feel I have a lot of energy in my game. I just want to win.”
Steur, who will wear the number 14 shirt, becomes Newcastle's third senior signing of the transfer window following the arrivals of French goalkeeper Ewen Jaouen and Ivorian winger Bazoumana Toure.
Newcastle begin their Premier League campaign at home to Liverpool on August 23.