Palace tabled a £1million offer for the highly-rated winger on Thursday, which was immediately knocked back.
An improved offer is believed to be in the pipeline, with Palace having begun their search for a long-term replacement for Manchester United-bound Wilfried Zaha, and a number of other clubs are also said to be keeping a close eye on Adomah's situation at Ashton Gate.
Adomah, who is currently away on international duty with Ghana at the Africa Cup of Nations, will not be pushing for a move, though, and is happy to leave a decision on his future with his club employers.
"It's not down to me whether I go," he said in the Bristol Post.
"At the moment, I'm happy at Bristol City. It's in the club's hands."
Adomah claims he was unaware that a bid had been made until asked about his future and will continue to leave the matter in the hands of his agent and City's football consultant Mark Ashton.
He added: "I have not heard of any request. I think they are letting me concentrate on the cup. I wish them luck with their game against Ipswich and, hopefully, they will do well."