Four high-profile signings - Argentine forward Mauro Zarate, Senegal midfielder Cheikhou Kouyate, highly-rated left-back Aaron Cresswell and promising midfielder Diego Poyet – have already been made, and Sky sources understand Ecuador striker Enner Valencia is lined up as the next arrival.
West Ham this week made a £12m bid for Valencia, who scored 18 times in 23 games for Mexican side Pachuca last season.
Sullivan has been working hard with manager Sam Allardyce to strengthen a squad that finished 13th in the Premier League last season.
"At the end of each season we always engage in discussion about which areas of the team need strengthening the most, before drawing up a list of targets to look at," the joint-chairman said.
"We have always preferred to get our key signings done early as you saw last summer with Andy Carroll and we have followed suit this year.
"The players we have signed so far were all high up on our list of priorities and I know they will all add something different to our squad - exciting attacking play, goal-scoring and experience as well as potential.
"The recruitment drive will carry on, though, as there are at least three more positions we would like to strengthen. We are in advanced talks to secure another one of our top targets and all being well we should get that deal over the line in the next 72 hours."
Sullivan also revealed he had received plenty of positive messages from supporters over their summer recruitment drive, which also included the return of Teddy Sheringham as part of their coaching staff.
"I have been really pleased with the correspondence I've received from fans over our transfer dealings so far,” he added.
“We were very clear at the end of last season that we wanted to provide more entertainment this season and the arrival of Teddy Sheringham as attacking coach and the four new players seems to have gone down really well.
"I think come kick-off in our first game against Spurs, we should have at least three more players so I'd like to think we'll go into the new season with a real sense of optimism about we can achieve."