The La Liga giants have been linked with a move for the Germany international several times over the past year.
Meanwhile, French outlet Le 10 Sport claims Barcelona are now weighing up a move for Gundogan as a replacement for Andres Iniesta.
However, the report states that Gundogan remains a key part of Pep Guardiola’s plans at the Emirates.
Iniesta left Barcelona over the summer to join Japanese side Vissel Kobe.
Gundogan was part of the Germany national team squad which crashed out of the 2018 World Cup following defeats to Mexico and South Korea.
Gundogan made just one appearance in the competition during Germany’s only win against Sweden
The 27-year-old provided seven assists and scored six goals in 49 appearances for Manchester City last season.
Manchester City take on Arsenal in their first game of the 2018/19 Premier League season on August 11.