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The winners of the two ties will meet in the semi-finals, while the other last-four clash pits France against Spain.
Kane was once again England's hero after leading the Three Lions to a hard-fought 3-2 victory over Mexico in the last 16 at the Estadio Azteca, despite playing much of the match with 10 men.
The Bayern Munich striker has scored six goals so far, helping England reach the quarter-finals.
Haaland, meanwhile, has thrived since arriving in the United States, scoring seven goals to inspire Norway to the furthest stage in the country's World Cup history.
That historic run, which included eliminating Brazil, has been accompanied by Norway supporters' trademark rowing celebrations, transforming stadiums across the United States into giant human Viking ships.
Norway coach Ståle Solbakken said on Friday: "Every match has been the biggest in the history of Norwegian football, especially in the knockout stages, so this is the third time we have played what is the biggest match of all."
Kane, 32, described his 25-year-old rival Haaland as "a beast", but stressed that they have different playing styles.
The England captain said: "I see myself as a different type of player, even though we score the same kind of goals."
He added: "I don't think we need to compare ourselves with each other. I have great respect for him as a player and as a professional. Of course, I hope he has a quiet day tomorrow, but his overall performances have been exceptional."
England received a major boost with the return of influential trio Declan Rice, Marc Guéhi and Reece James to training on Friday after illness and injury.
Head coach Thomas Tuchel described their return as "the best news", adding: "We took a big step in our last match, but it was only one step."
The German continued: "The next step is to win the quarter-final. The most important thing is to focus on what lies ahead."
The match at Hard Rock Stadium in Miami is expected to be played in hot and humid conditions, with temperatures forecast to exceed 40°C.
Argentina began their title defence in convincing fashion as they seek to become the first team since Brazil in 1962 to retain the World Cup.
However, as the tournament has moved into the knockout rounds, signs of vulnerability have begun to emerge after difficult encounters with World Cup debutants Cape Verde and then Egypt.
Lionel Messi, who celebrated his 39th birthday during the group stage, has once again been Argentina's inspiration, repeatedly rescuing his side with moments of brilliance.
Even so, Switzerland, renowned for their organisation under veteran captain Granit Xhaka, may be capable of exploiting any signs of fatigue among the Argentines and causing problems at Arrowhead Stadium in Kansas City.
Argentina coach Lionel Scaloni said on Friday: "They have very experienced players and they are a strong team, so they will certainly be difficult opponents."
Messi will be aiming to extend his record tally of 21 World Cup goals, an unprecedented total accumulated across his six appearances at the tournament.
Spain secured a 2-1 victory over Belgium with a late goal to book a blockbuster semi-final against France.