Scott Murray and Enoch Showunmi earned the Robins a deserved third round replay win over the Championship side at the Ricoh Arena, prompting the home fans to call for the head of Adams.
"This is a time not to be worrying too much about myself, because all will be revealed." said Adams, who then refused to confirm whether he had already been told to expect his P45.
"When I say that I mean I will not be resigning. But I am not going to go into all that now, because it will all come out anyway - and I think that would be disrespectful to Bristol City.
"They should have the credit they deserve because it's their night, they came here full of confidence and had belief in what they were doing.
"We lacked that, and there was only one winner. We can only apologise to the supporters for a performance which was not good enough to say the least. We got what we deserved and we should be out of the cup."
City's reward is a fourth round clash with Premiership Middlesbrough, who beat Hull City 4-3 in an action-packed replay at the Riverside Stadium, at home.