McGovern had an exceptional competition, producing the goalkeeping performance of the finals against World champions Germany.
On Saturday in the 1-0 defeat to Wales, he was only beaten by an unlucky own goal from Gareth McAuley, and became the first number one in the tournament to stop Gareth Bale from scoring, saving a dangerous free-kick from the Real Madrid superstar in the process.
The 31-year-old is out of contract at Hamilton at the end of the month and already has had much interest from England and Scotland and there have been suggestions that German clubs may be keen.
"The tournament's been brilliant for me personally. All the boys would say the same, it's been an amazing experience, the best I've had in football," said McGovern.
"I will cherish the memories forever and I know my family will too.
"Regarding my future, I don't know what's happening because I've been focusing on the Euros. The games come so thick and fast it's hard to focus on them, never mind the future. This week I'll sit down and hopefully I will have some good options.
"There's been a bit of interest since the Germany game, as there was before the tournament. I wanted to come to France without signing a contract because I thought it was a great opportunity. Obviously Hamilton are in touch as well.
"I've backed myself. It was a bit of a gamble, but I backed my ability and it's turned out okay. If you have confidence in yourself, it shows a bit of character."
McGovern, a modest man by nature, has been taken aback by all the praise he has received and feels Northern Ireland are desperately unfortunate not to still be in the tournament.
He said: "I don't think we deserved to lose the game against Wales. We had the better chances. Bale had the free-kick, and then there was the goal, but I think we kept him quiet. Gareth McAuley couldn't do too much with the goal and that proved to be the difference."
Striker Kyle Lafferty added: "I don't think they even had a corner. We were the better team, but we're going home disappointed. We can hold our heads high after that massive performance.
"We knew that we could win the game and to go out and put in the performance we did on the pitch, we're really gutted to be out.
"They got lucky. They will know that they were second to every ball.
"All the players are proud to play for the shirt and hopefully all the fans have seen that. Our supporters have been brilliant throughout the tournament."
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