Burnley sign Gudmundsson and Pope from Charlton
Burnley did not disclose financial details of the deals, but both players have signed three-year contracts, with the club having the option to extend for an additional year.
The 25-year-old Gudmundsson, who had a joint league high 11 assists in the Championship (second-tier) last season, started every one of his country's games at Euro 2016.
Iceland pulled off the upset of the tournament when they defeated England in the last 16, before being eliminated by hosts France in the quarter-finals.
"Johann... is a quality player who we saw last season playing at Charlton in a tough time and we liked the quality he showed," Burnley manager Sean Dyche told the club's website. (www.burnleyfootballclub.com)
"It helps to see a player in the European Championship with that kind of company and he did well there so it was one we all agreed would be a good acquisition for the club and we look forward to working with him going forward."
Pope, who made 28 appearances as Charlton were relegated to League One last season, will compete with Tom Heaton and Paul Robinson for a place in Dyche's starting XI.
Both Heaton and Robinson are former England internationals and Pope said the chance to train alongside his compatriots was key to him making the switch.
"I want to develop as a goalkeeper and with the keepers that are already here I definitely feel I can learn from them," the 24-year-old added.
"They've both experienced the England set up and it's only a good thing for me to be around them day to day in training and at games. I'm really looking forward to it."