With their remaining fixtures in mind, there's really not a good time to sack him either, to make the players believe the new manager when he says they can turn it around.
Rest of the season for them:
Manchester United (H)
Leicester (H)
Liverpool (A)
Chelsea (H)
Leicester (A) (yes, really. The match 3 rounds earlier is a postponed round 18 match)
Watford (A)
Brentford (H)
Crystal Palace (H)
Arsenal (A)
Do they sack him before the weekend now? Or let him play us, Leicester and pool? Unless he wins 2 of them (or 1 win 2 draw?) he gets sacked before Chelsea. Bring in someone like Big Sam for the last 6 games to make a stand and grind out results to keep them over the line?