Psyren, PSYЯEN (pronounced Siren) is a 145 chapter Manga by Iwashiro Toshiaki, that I really enjoyed reading., I will also attempt to not spoil much of the story.
Basic Plot The story follows Ageha Yoshina, a high-school boy who will help anybody with their problems for a small fee of 10,000 Yen. After hearing a pay-phone ringing, he picks up the phone, instead of someone else on the other end, Ageha receives a phone-card with the word "PSYЯEN" printed on it. A female classmate disappears after mentioning PSYЯEN, and Ageha uses the phone-card in hopes of finding her. After answering a lengthy survey over the phone he is asked if he wants to go to Psyren. The next day Ageha hears a phone ringing, when he picks up a phone, he is transported to the world of PSYЯEN.
After finding the classmate that dissapeared, Sakurako Amamiya, Ageha learns that this world is inhabited by creatures called Tavoo and they were brought there by a mysterious person called "Nemesis Q". Amamiya, having been there many times before tells Ageha what he must do to get back home. Once a person has been to PSYЯEN, they awaken special powers know as PSY.
Ageha must learn the secrets of PSYЯEN in order to change the future, and to stop the constant calling.
Final Thoughts: I loved this series, and I wonder why it hasn't been made into an anime yet. The characters were developed rather nicely, and the art style is probably one of the more unique ones that I have seen. All-in-all I will give it 9 PSYЯEN Calling-Cards out of 10.
What you are thinking about is not an actual game. In the manga, when they are transported to PSYREN, they have a mission to go to a phonebooth to be returned to the present time. Many characters in the manga view this as a game and might be where you thought it had a game.