And Real Madrid coach Ancelotti, himself under serious pressure at the Bernabeu, is one of the names being considered at Old Trafford.
Real have just suffered back-to-back defeats in La Liga and this week face Borussia Dortmund in the last eight of the Champions League.
If the Spanish giants were to go out to the Bundesliga club – for a second season running – it would leave Italian Ancelotti heading out.
Klopp, along with Atletico Madrid coach Diego Simeone, is also in the running to replace Moyes.
The derby crushing at home to Man City again raised big concerns about the Scot’s ability to motivate the United dressing room.
Should they suffer another pasting in front of their increasingly unhappy fans on Tuesday, when they host Bayern Munich, it will be the final nail in Moyes’ coffin.
Ancelotti, 54, is deemed a far safer appointment than Klopp.
In his first season as Chelsea boss, Ancelotti led the Blues to a first domestic League and FA Cup Double.
Ancelotti could also act as the senior managerial figure while some of United’s class of ’92 – Ryan Giggs, Paul Scholes and Phil Neville – prove their worth as coaches.