The 19-year-old has attracted considerable attention during his first season in Swedish football after joining Häcken from Ivorian club ASEC Mimosas at the beginning of the year.
Doumbia signed a contract running until the end of the 2030 season, but his impressive development has already prompted interest from several clubs across Europe.
Häcken sporting director Erik Friberg has acknowledged the growing attention surrounding the midfielder, although he stressed that there has yet to be a concrete proposal.
“There is a lot of interest when you look at which clubs have been here to watch him and which clubs are talking about him. Right now there is nothing particularly concrete, but I think that could happen eventually,” Friberg told Göteborgs-Posten.
The list of clubs reportedly monitoring Doumbia is extensive. Trabzonspor, Basel, Nice, Strasbourg, Rennes, Sunderland, Coventry City, Celtic, Paris FC, Bayer Leverkusen, Club Brugge and Freiburg have all been linked with the midfielder.
However, Everton are currently ahead of fellow English clubs Sunderland and Coventry in the pursuit of his signature.
Moyes is understood to be prioritising midfield reinforcements, with Doumbia emerging as an interesting option following his performances in the Allsvenskan.
The teenager has registered two goals and three assists in 26 league appearances during his debut campaign in Sweden, quickly establishing himself as an important member of Jens Gustafsson's side.
Häcken are fourth in the Allsvenskan standings after 28 matches, three points behind third-placed Djurgårdens IF.
With Doumbia under contract for another four years, Häcken are in a strong position when it comes to determining his future. Nevertheless, the growing interest from the Premier League and other European competitions suggests the club could face an increasingly difficult task retaining the midfielder.
For Everton, the coming months could provide an opportunity to pursue a player whose development has attracted attention well beyond Swedish football.