Despite conceding that the aim could be out of reach for Sam Allardyce's side, Sullivan believes that it is only right to set high targets for the Hammers.
"You can't just look to stay up," the Welshman, who co-owns the club with David Gold, told The Mirror. "It is not acceptable to our supporters to say, 'We've survived. We've had a very good campaign.'
"We are not Norwich. We are not Fulham. We are not Reading. We are West Ham United.
"Our supporters are realistic. They know it is a huge and difficult task to break into the top six. But we must be aiming for that, even if we don't achieve it."
West Ham finished 13th last season.