The Ghana international was expected to attract interest this summer following the Robins' relegation out of the Championship in 2012/13.
The 25-year-old impressed during three years at Ashton Gate in the second tier and is likely to he offered a route back up the League ladder.
City are, however, reluctant sellers and will only enter into transfer talks if the terms are right for all concerned.
That is not the case at present, with Sky Sports of the understanding that Wigan Athletic are the side to have seen their advances rebuffed.
"Bristol City Football Club can confirm it has received two undisclosed bids from an unnamed club for winger Albert Adomah," read a statement released by the Robins.
"Both offers have been rejected.
"The club plans to make no further comment at this time."