Gallagher left Chelsea a year ago to join Atletico Madrid, where he had a successful first season in Spain.
Atletico Madrid's coach, Diego Simeone, is reportedly pleased with Gallagher's performance and attitude, but the club is open to selling him for a price of around €45-50 million.
Newcastle United's manager, Eddie Howe, is said to be an admirer of Gallagher and has suggested him as a potential new signing to the club's chiefs.
Tottenham Hotspur are also interested in bringing Gallagher to the Premier League, but a transfer could be complicated due to his background at Chelsea.