The competition's least populous nation are appearing at their first major tournament, and qualified from Group F with two draws and a win.
Now they have been drawn against England in the last 16, and Sigurdsson, who has spent the majority of his career in the Premier League, cannot wait for the clash in Nice.
"I have dreamt of this since I was a kid," he said ahead of the game. "Playing against England and doing so in the round of 16 is huge. We are full of excitement.
"We were so close to winning in the second game, but we messed up. That's why we were really relieved to beat Austria and qualify. You dream of celebrating with your fans like that."
"England were not at their best in group stages, but still qualified - it illustrates how strong their squad is," he added. "They have some great attackers, who have been scoring goals for fun last season.
"I am looking forward to playing against my former team-mates. I am good friends with the Tottenham players. We are still in touch and have been sending each other text messages."
However, Sigurdsson insisted that he would leave the game plan to the coaching staff.
"It is a very important game for the team and the country," he said.
"All the players will know England. I won't be giving tips - the coach will come up with a plan.
"We are looking forward to the game, it will be very exciting."