Iceland, appearing at their first major tournament, reached the quarter-finals of Euro 2016 with a sensational 2-1 win over England on Monday.
Their charge to the last eight has drawn comparisons to the surprise exploits of Leicester, who shocked the football world by claiming a maiden Premier League title in 2015-16.
The two places have a similar population - around 330,000 - and joint-coach Hallgrimsson admits he does see comparisons between Iceland and the English champions, and would love to see a similar story play out in France.
"I really would like things to end like they did with Leicester," said Hallgrimsson at a media conference on Wednesday.
"They played to their strengths and we are trying to do the same. I think there is the same kind of spirit in both teams, we are willing to work together."
After falling behind to a fourth-minute Wayne Rooney penalty in the last-16 clash with England, Iceland hit back quickly with goals from Ragnar Sigurdsson and Kolbeinn Sigthorsson.
Hallgrimsson was seen laughing alongside joint-coach Lars Lagerback following Sigthorsson's strike, and he revealed the reaction was prompted by a feeling of relief.
"It was just a moment of relief. We said before the match that conceding an early goal was the worst possible thing," he continued.
"To answer that quickly was a psychological boost for us. It shocked us a bit when they scored this early, but I think it shocked the English even more to concede right away. There was nothing special about that laugh, just in the moment, it was just fun."
Iceland will take on hosts France in the quarter-finals, but regardless of the outcome Hallgrimsson is confident the profile of the Nordic nation's game has benefited.
"I think the attention has changed, I've said it before this has put Icelandic football on a higher standard," he added.
"It has not only changed things for the players here, but for all Icelandic players. This should be to the benefit of Icelandic football as a whole.
"To beat England will always be in the minds of Icelanders, the players will be heroes back home."
Hallgrimsson will take sole control of the team when Lagerback steps down at the end of the campaign.
The 49-year-old was effusive in his praise of the former Sweden boss, who he feels has played a hugely influential role in the development of Icelandic football.
"You have to realise that Iceland is amateur football," said Hallgrimsson.
"So to have a guy coming to Iceland with all of his international experience, someone who has worked on this level for so long, is priceless.
"He is leaving all his knowledge in Iceland and that is his legacy. I know he doesn't like compliments but it's priceless."