The 63-year-old also reacted angrily to accusations that he did not take the FA Cup seriously following his side's defeat to Blackburn Rovers.
"You have your ups and downs and we are having a bit of a down," Hill-Wood told the Daily Star. "But some of the reaction has been a bit hysterical. You hear clubs being criticised for sacking their manager too quickly. Now there are people out there calling for us to sack a manager who has been a huge success for 15 years.
"That does not help much. I have been extremely disappointed by some of the recent results. I have been an Arsenal fan for a lot longer than most so I think I know what it feels like. It hurts a lot and I am sure Arsene is hurting too.
"He is very diplomatic and calm but deep down he feels very strongly and he is as upset as anyone. It's not just me who is backing Arsene. Stan Kroenke is very supportive too."
The Gunners are due to face Bayern Munich at the Emirates this evening for the Champions League round-of-16 first leg.