Tom Ricketts has said that his family and bid team have an "absolute commitment" to diversity and inclusion if they succeed in buying Chelsea.
But there are concerns over the bid after it emerged that the family's patriarch, Joe Ricketts, was accused of Islamophobia three years ago.
The Chelsea Supporters' Trust (CST) said it is against the family's bid.
The CST said it had conducted a snap survey of its membership on 1 April to see if "they had confidence that the Ricketts family would run an inclusive and successful club" and "whether they supported" the bid.