The Reds currently maintain a four-point lead over Manchester City as the two look set to battle in one of the most exhilerating title races in recent memory.
Gillespie was a part of the Reds' last title win in 1990 and won three league crowns while with Liverpool.
The club has yet to capture an ever-elusive Premier League crown, though, and this group has a chance to finally seal what would be a special achievement for Liverpool.
“The longer it goes on without Liverpool winning the title, then the team that last won it becomes better and better, and our reputations grow,” Gillespie told the Mirror. “That’s nice from our point of view, but I can tell you every single player from 1990 wants nothing more than this team to win the title again.
"We know what it means, for the fans and for the club, and we know how it feels. It elevates you if you are successful. There are players from my era who are legends because of that, because they were serial winners, and I would say this team would get the same from the fans.
“It is a chance to become legends.”
Gillespie admitted that he never imagined the Reds having to wait this long to return the glory having witnesses the dominance during the 1970s and 80s.
However, he does see some similarities between his title-winning sides and the current Reds squad.
“There are comparisons between the squads – they are pretty much the same in many regards,” the former Scotland international said.
“There is a real strength in depth in Jurgen Klopp’s squad, and we had that, a quality running throughout it, even outside the first 11 players. There seems to be a belief and confidence too, not just in individuals, but in each other.
“I look at them and really do think they have got strong enough characters in there, like we had, to do it... and now would be as good a time as any to break that spell without the title.”
With Manchester City hot on their tails, though, the Reds will face a difficult challenge.
This past weekend's 4-3 win over Crystal Palace was a notable test, and there will be plenty more along the way.
Next up for Liverpool is a clash with Leicester City next Wednesday as the team looks to maintain its lead atop the league following a break from action.
“Every game now there is an unbelievable expectation and pressure. But that is where the work of the manager and staff comes in, and the building they have been doing, even in pre-season," Gillespie said.
“We went away in the summer for a tour, and it was more team-building that results based. They are doing the same going to Dubai now for a break. It brings them together. It makes them believe in each other. From my experience, when it comes down to it, there will be moments when you have to trust your team-mates completely.
“Klopp’s team have that — they believe in each other. We lost a few games that season, but not the belief we had. That will be the most important thing.”