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BBC Radio Bristol understands the bid, from an unnamed club reported to be Birmingham City, totalled £4m.
"He has definitely had his head turned, there is no question of doubt about that," Johnson said of Flint, 28, who signed from Swindon Town in 2013.
"We don't have to do anything that we do not feel is right for Bristol City."
Flint has scored 29 goals in 200 appearances for City, while 23-year-old Bryan, a graduate of the Robins' academy, has played 181 times since his debut in 2012.
"We a received a bid yesterday [Tuesday] and I am not sure of the exact time, but I know one minute later it became public, which is very, very disappointing," said Johnson.
"It is a derisory bid in my eyes and the club's eyes. It is a great place to be at the moment, Bristol City, because we are in a really good place with Financial Fair Play. Nothing needs to be forced upon us by anybody.
"Both players have got good, strong healthy-length contracts. If anybody is to go, it will be on our terms because we have a good owner and we are to trying to bring players in and certainly not trying to lose our best players."
Meanwhile, City have named Australia defender Bailey Wright as their captain for the new season.
Bristol City have rejected a "derisory" offer for centre-half Aden Flint and left-back Joe Bryan, according to head coach Lee Johnson.