Morgan was replaced at half-time in Saturday's 1-1 draw with Aston Villa through injury and was absent from Monday's training session at the City Football Academy.
The boss explained that a decision on whether the defender would be fit for Wednesday's trip to the Vixens would be made in the next 24 hours.
Taylor also suspects that the Ashton Gate clash - set to get under way at 19:00 (UK) - might come too soon for Stokes to make her return to the fold, although he did reveal that she is 'improving every day'.
“We need to monitor her over the next 24 hours," the boss said of Morgan.
“She didn’t train today, so at this moment in time she looks a doubt which is unfortunate because she’s an important player, like all the players are.
“We’ll have to assess her over the next 24 hours and see if she’s able to make the journey to Bristol.
“Demi is improving day by day, but this game might be a little too soon for her.
“Outside of that everyone has come through and trained today as normal.”
City come into the tie on the back of a 1-1 draw with Aston Villa in the Barclays Women's Super League.
Although Taylor's side were able to extend our unbeaten run in all competitions to 13 matches, the boss reflected that his side didn't perform to our usual high standards.
However, he has been impressed with how his players have responded on the training ground ahead of our return to Conti Cup action.
The boss said: “[The reaction has been] good. If we get most of the elements of our play right, the score takes care of itself.
“It doesn’t guarantee you the win, but that was the disappointing part for me [against Aston Villa].
"We just didn’t play our normal game against a decent Aston Villa team who are completely different in terms of personnel this season and have some experienced players in there.
“A draw was probably a fair result. We knew we kind of let ourselves down a bit on the performance and that’s frustrating for all of us.
“[But] We had a good honest conversation this morning about areas where we can improve and the attitude in training has been top class.
“Now we look forward to two different competitions in the space of a few days.”