The American duo require at least 75% ownership to assume unconditional control of Liverpool.
Formal offers were sent to shareholders nine days ago.
There are still 12 days left to decide whether to sell stakes in the club, but the initial response from shareholders seems positive as they seek to protect benefits such as priority ticketing.
There are not expected to be any problems hitting the 75% target which is conditional if the Gillett/Hicks bid is to go ahead.
More tricky could be acquiring the 90% for full ownership, which depends on the sale by shareholders in possession of a larger stake.
If the U.S. pair buy 90% of the club, the remaining 10% passes to them on a compulsory purchase.
Chief executive Rick Parry expects developments to proceed over the next few days, and is hopeful the 75% mark may even be reached in time for the new owners to attend this weekend's Manchester United clash.
"There's no feedback yet on the percentage of shareholders who will sell, but we may have an idea by later this week," said Parry.
"We're obviously in a slight limbo period at the moment while we wait and see.
"The feedback I've had in terms of letters or e-mails, which I requested for those with queries, has been extremely positive, which is encouraging. The vibes I've had from shareholders have been really good.
"Maybe we'll have a clearer idea what stage we're at by the end of this week or early into next week.
"The shareholders have 21 days from the date when the offer went out. At this stage we're just nine days into that period. If an early decision is taken, that's fine, but we appreciate it's a big a decision for every shareholder and they have to consider what they think is best.
"You don't have to wait the full 21 days to reach the 75%, so there may be an idea earlier than that of the situation. From the club's point of view, the sooner the better.
"Deadlines are imminent with the stadium and it's important to get on site, get the preparatory work done and keep the ball rolling. We don't want the interim period to slow that process down."
Parry spent time in the United States two weeks ago for further discussion on the club's future with the new owners, and remains excited by the long-term prospects under their chairmanship.
"There was a lot of brainstorming and good ideas to come out of that," said Parry.
"There's no intention of copying everything they do in America because it's different. The nature of the sport is different, the nature of people who attend events is different and the culture is different.
"But, undoubtedly, there are things they do phenomenally well which we can learn from. The quality of arenas in the states is by and large truly inspiring.
"The attitude of the new owners is so far very much one of telling us what they can do and saying if we think it's of benefit, do it, but if it's not right for the club, don't. That's encouraging for us.
"You have to be open minded and go into this with the attitude of hearing good ideas and thinking how much we can benefit.
"At the same time, there's tremendous respect for the values we have here. We're not going to be importing cheerleaders or adopting things we know would be a nightmare to our fans, so there's no need to worry about that.
"It's a balance of being modernising without undermining the traditions which really matter. It's a genuinely exciting time. They've a lot of energy and good ideas.
"There will be a lot of change, but I wouldn't see it as a revolution. It's about preserving the things which are good and building upon them. As ever, the first priority is success on the pitch. That unites everybody."
Both Gillett and Hicks hope to attend this weekend's clash with Manchester United and the Champions League clash with Barcelona, but that's still to be confirmed.
There are also plans for Gillett to engage in a dialogue with supporters' groups.
"It wouldn't surprise me if George took himself to The Albert for a drink with the fans," said Parry.
"That's the kind of thing he's done in Montreal. He will speak to the fans. He's a people person with a thirst for knowledge and he's a very good listener.
"The reason George and Tom have kept away from games so far is out of respect for the shareholders.
"They don't want to appear presumptious because they recognise it's big decision being taken and shareholders should be given the time and space to make it."