Hart was also left on the bench for City's League Cup semi-final, first leg at Everton on Wednesday and Argentine Willy Caballero will continue in his place on Saturday.
"Willy will play both cups. For Joe it's good to work one complete week because we are playing so many games," Pellegrini said at his news conference.
Captain Vincent Kompany is recovering from another calf injury while fellow defender Eliaquim Mangala and striker Wilfried Bony are also unavailable.
"Vincent is progressing, it's a muscle injury and we must not rush him," Pellegrini said.
City are still in all four competitions which they started, although they must retrieve a 2-1 deficit against Everton.
Pellegrini says a strong FA Cup run is important, even if that would place extra burdens on his squad.
"I think one of most important merits of this squad is to continue being in all the competitions with so many injuries," he said. "The priority for my team is to try and win every game.
"It doesn't matter which competition you are in. That gives you an ambitious mind and a winning mind."
(Reporting by Martyn Herman; editing by Toby Davis)
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