The MCC will begin the final day in Abu Dhabi needing 176 runs to claim victory with eight wickets in hand.
Stoneman hit 115 but it was a near lone hand as Durham were bowled out for 257 yesterday.
The MCC were 48 for two at the close after openers Sam Robson and Luis Reece were unable to negotiate the final 20 overs of the day.
Daniel Bell-Drummond and Virender Sehwag will resume the MCC’s chase on the final morning.
Defending county champions Durham began the day looking to build a potentially match-winning lead but, aside from Stoneman, only Gordon Muchall (56) passed 20.
Nottinghamshire left-armer Harry Gurney claimed three for 50.
Jamie Harrison and Scott Borthwick kept Durham in the game as they took a wicket apiece before stumps, although the MCC were still well placed.
South Africa have been penalised for a slow over-rate in Monday’s narrow win over New Zealand.
The Proteas, who won the World Twenty20 Super 10 match by two runs thanks to a stunning display of death bowling by Dale Steyn, were judged by to have fallen behind the required rate by two overs.
Referee Javagal Srinath imposed a fine of 40 per cent of captain Faf du Plessis’ match fee as well as half that for each of his team-mates. Du Plessis faces a one-game suspension should he fall foul of a similar offence in the next 12 months.