Senior officials at the Upton Park club announced earlier this week that they expect a minimum top-10 place along with a better style of football from the 59-year-old's side in 2014-15, but Sullivan has raised expectations even further.
Sullivan, who bought control of West Ham in 2010 alongside David Gold, has called on West Ham's players to challenge for silverware and push for Europe.
"We start next season with three competitions, two of which we can win and we have to be dreaming of winning the third one," he is quoted as saying in the Evening Standard.
"I don't want to just aim for top 10. We should be aiming for top six.
"I want to be looking down the table not up it next season."
And Sullivan revealed the club are in advanced negotiations with a number of players that he hopes will improve the West Ham squad next season.
"We have got five or six players lined up," he added.
"Most are young players, the youngest is 19 but has already played top-level football in two countries, while the oldest is 27.
"We want to see players who can play in any of the three attacking positions and who have a proven record of scoring goals.
"We want people who take on players and go past them, people with movement because we need to add speed to our team."
