BBC writes that Arsenal has decided to bet on Gyökeres and the negotiations with Sporting Lissabon are far advanced.
Arsenal has finished second in Premier League two seasons in a row and just the lack of a sharp striker decided the club's fate the past season.
The Athletic writes that Gyökeres, who made 54 goals in 52 matches for Sporting the past season, has been offered a five-year contract.
According to the Portuguese sports newspaper A Bola, the 27-year-old Swedish national team striker does not need to join Sporting on Monday when the club's preseason training is to start, but has been given permission to join on Friday if the deal is not completed by then.
Viktor Gyökeres has previously also been linked to Manchester United and Juventus, but now Arsenal seems to be the first priority.
Aftonbladet writes that Viktor Gyökeres will be the most expensive Swedish player deal so far and beats both Zlatan Ibrahimovic's transfer from Inter to Barcelona and Alexander Isak's move from Real Sociedad to Newcastle, which both ended up at around 750 million kronor.