Minasyan's side picked up several impressive results in Group B, including wins in the Czech Republic and Denmark, crushing Morten Olsen's side 4-0.
They also ended the group phase with a creditable 2-2 draw against Italy in Naples, but just one win and four defeats from their home fixtures destroyed their chances of reaching next year's tournament in Brazil.
He said: "I prefer another continuation to my career. There are several offers, specifically from abroad, and I want to discuss them.
"I would like to thank Ruben Hayrapetyan, who entrusted me the duties of the national team head coach for the past four years. It is a great honour for every coach to be the manager of a national squad.
"I also would like to thank our football fans. I have always felt their support. I am confident that they will always be with our national team, our football.
"I also would like to thank the media representatives. I have always been honest with them.
"And most importantly, I want to thank our players. It has been a great pleasure for me to work with them. Together we passed along a road which, I believe, was useful. They have a great potential, they are professionals, and I am confident that they still will climb great heights."
