England's 0-0 draw with Slovakia saw them progress from Group B as runners-up, and they will subsequently face Iceland in the last-16 after the Nordic newcomers defeated Austria 2-1 in their final Group F clash.
The Three Lions have struggled to score so far at the European Championship, while Hallgrimsson and Lagerback's debutants have proven to be a stubborn, defensively organised opponent.
That raises the spectre of yet another dreaded shootout to decide who proceeds to the quarter-finals and, perhaps not fully appreciating England's awful record from the spot on the big stage, Hallgrimsson sounded a note of surprise when asked if his players will be practicing penalties this week.
"No we haven't, but probably we'll start it now. I [already] had a question [about it]. You are obsessed with penalties in England," he said.
"I had a question from an English guy there and he asked me, 'what's the ratio?'. And when I think back we've scored every [time]. We always score our penalties. So watch out."
If Iceland's penalty prowess is not worrying enough, Lagerback has an impressive record in competitive matches against the Three Lions, securing draws against the more fancied heavyweight side with Sweden at the World Cup in 2002 and 2006.
Lagerback said: "It's always nice playing England, I have very good experiences of that. We will try to keep that going."
Lagerback will be going up against a man held in high esteem in his native Sweden, England manager Hodgson having, along with Bob Houghton, helped to modernise football in the country.
"I've known Roy since the 70s," he added.
"So that gives me a little bit of an extra kick. [But] I won't be taking any contact from him, he has his hands full like we do. He's one of my friends, I regard him as that."