The 25-year-old, who represented Ghana at this year's Africa Cup of Nations, was the subject of two bids from an unnamed club - believed to be Wigan - last week.
City revealed that they had turned down both offers and have also released a statement saying that Adomah will only be sold if they receive an acceptable offer.
"Bristol City can confirm that Albert Adomah has submitted a written transfer request," read the statement.
"As far as the club is concerned its position remains unchanged.
"As with any player, until an acceptable offer is received he will remain a Bristol City player.
"The club will be making no further comment at this time."
Adomah, who moved to Ashton Gate from Barnet in 2010, made 41 appearances for City last season, scoring seven goals, as the Robins were relegated to League One.
His current deal runs out next summer.