He told the Hull Daily Mail: "There's no point in playing either of them half-cooked against Leeds and then seeing them ruled out for the rest of Christmas as a result.
"I don't think the physio would allow me to play them anyway because he's a professional in his own right and because of that, he will direct me to make the right decision.
"If he thinks it's a risk and I think that it's a risk, there is no way we will play either Nick or Jon.
"Jon's in exactly the same boat as Nick and we'll give it right up until the last minute for them to show they are fit."
Bridges replaced the injured Barmby midway through the second half against Cardiff on Saturday, and came on to score City's fourth goal.
Brown is unlikely to make any further changes to the personnel or the formation that enjoyed great success against the Bluebirds.
Expected line-up (4-3-3): Boaz Myhill; Sam Ricketts, Danny Coles, Damien Delaney, Andy Dawson; Dean Marney, Ian Ashbee, David Livermore; Steve McPhee, Michael Bridges, Craig Fagan.
Subs: Matt Duke, Michael Turner, Ryan France, Stuart Elliott, Nick Barmby/Jon Parkin.