Avram Grant's reign as manager has begun with just one win in 10 games, leaving his side propping up the Premier League.
Sullivan returns to former club Birmingham on Saturday - though he will not be accompanied by colleague David Gold, banned from St Andrew's by Blues owner Carson Yeung - and the Hammers then face promoted sides West Brom and Blackpool.
Liverpool, rebuilding after a poor start to the season, follow at Anfield and Sullivan, in remarks reported by the Daily Mail, said: "I'm looking for at least seven points from the match at Birmingham followed by West Brom and Blackpool next week and then Liverpool away.
"If we don't achieve that goal, we know we have a real problem and that we have to shuffle the pack."
In a separate interview with the Times, however, Sullivan insisted those changes are unlikely to impact on Grant's position in the near future.
"There have been unhelpful stories and (there is) not an ounce of truth in them, " he said. "We have not looked at anyone else and have supported the manager. He is very confident.
"We went through tough times (at Birmingham) with Steve Bruce and Alex McLeish. We have learnt from that and that is what is confusing us."
He admitted the extent of the Hammers' problems, though, saying: "We thought we had done enough to have a team at least 10 points clear of relegation. I can't quite get my head around it.
"That is the disappointment, although we feel we have not played too badly. You wonder if your judgement has gone. We are quite happy with what we have seen, apart from results. We hope that we can turn it around."