Arsenal have completed the permanent signings of Martin Odegaard from Real Madrid and goalkeeper Aaron Ramsdale from Sheffield United.

Norway international Odegaard spent the second half of last season on loan at the Emirates Stadium, scoring two goals in 20 appearances for the north London club.

The 22-year-old became the Gunners' fourth summer signing after the arrivals Ben White, Nuno Tavares and Albert Sambi Lokonga, before the North Londoners confirmed Ramsdale as their fifth arrival on Friday evening.

Former England Under-21 international Ramsdale was a major target for Arsenal manager Mikel Arteta but the London club was initially put off by Sheffield United's steep asking price, reportedly worth £40 million.

"The club turned down a number of bids from Arsenal and expressed a desire to keep Ramsdale at Bramall Lane for the long-term," Sheffield United said in a statement.

"However, discussions earlier this week with the player and his representatives highlighted his desire to further his career away from S2.

Martin Odegaard spent the second half of last season on loan at Arsenal

"In addition, Arsenal returned with another improved, head-turning bid. It was felt the best decision was to accept a club record."

Ramsdale, who joined Sheffield United from Bournemouth in August 2020, played all 38 games in the league last season, keeping five clean sheets.

He was named Player of the Year at Bramall Lane but was unable to prevent the club from being relegated to the second-tier Championship.

Ramsdale's arrival at the Emirates Stadium is likely to see second-choice keeper Alex Runarsson, signed from French side Dijon last year, leave the club.

Odegaard joined Madrid as a 16-year-old in January 2015 from Stromsgodset in his homeland and later spent two seasons on loan at Heerenveen, before temporary stints with Vitesse Arnhem, Real Sociedad and Arsenal.

He will not be available for the Gunners' Premier League clash with Chelsea on Sunday as Mikel Arteta's side look to bounce back from the opening day defeat to Brentford.