"Our family is extraordinarily pleased with him, we think he is absolutely as good as there is in the business and I am sure the Hicks's feel the same way," Gillett told the club website on Thursday.
Booed off after Tuesday's 2-1 home Champions League defeat by French club Olympique Lyon, Benitez's side have now lost four successive games in all competitions and trail league leaders United by seven points.
"I think the run of results disappoints everybody. Certainly, it disappoints the fans and it disappoints Rafa... but we are in this together," Gillett said.
Former Valencia boss Benitez, whose new deal signed in March lasts until 2014, has not been helped by injuries to key players Fernando Torres and captain Steven Gerrard and both are doubtful for Sunday's clash.
Since thrashing Hull City 6-1 last month Liverpool have scored just once and conceded seven goals.
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