Leicester have elevated themselves into the top-four after making a brilliant start to the new season.
At the moment, the Foxes are level on points with top of the table Manchester City, and Savage believes they have a great opportunity.
"You might see Leicester's ambition. If Leicester think they can finish in the top six, let's see who they can attract in January," the former Foxes midfielder said.
"Because now they're not attracting players for staying up, this season they're going to attract players to maybe qualify for Europe. That's unbelievable to think.
"So if they can keep hold of Riyad Mahrez and Jamie Vardy, and add two or three very, very good players, they've got a good opportunity."
Leicester may be keen to bolster thank ranks in the January transfer window, but they will also be wary of unsettling their current group.
The Foxes have built the foundations of their success on the team spirit created at the King Power Stadium, and it will be crucial that Claudio Ranieri does not disrupt this by bringing in too many fresh faces.
Showing some intent will be crucial for Leicester, though, as they do have a genuine chance to claim a European spot this term.
The Midlands outfit have only lost one game all season so far, and the quality of their attacking play means that they have been a threat to every side they have faced.
Leicester’s next fixture comes against struggling Swansea City, before they face Chelsea, Everton, Liverpool and Manchester City over a demanding Christmas period.