The combative former Holland midfielder, whose illustrious career has seen him play for Ajax, AC Milan, Juventus, Barcelona, Inter Milan and Tottenham, is available on a free transfer after his second spell at Ajax came to an end in the summer of 2008.
Mandaric, whose manager Nigel Pearson is among the early favourites to replace Gareth Southgate at Middlesbrough after the Teesiders sacked the former England defender on Tuesday night, believes signing a player of Davids' calibre could be pivotal to the club's ambitions of returning to the Premier League.
And he is hopeful of getting a decision from the 36-year-old before the end of the week.
"If we couldn't afford him we wouldn't be talking to him," Mandaric told Sky Sports News.
"We are a very ambitious club but there is always room for improvement and our goal is to really give it a go and try to get up this year and players of that kind of profile will help us.
"We are in conversation with Edgar and his agent, and at this point that's all I can say. We are progressing and it is potentially exciting, of course, but we are not there yet.
"I think things will be taking place very shortly and not more than a couple of days."
Mandaric's former club Portsmouth are also interested in signing Davids, who has not played competitive football since leaving Ajax almost 18 months ago.
Mandaric expects competition for Davids' signature and does not expect the player's age or fitness after so long out of the game to be a problem.
"I don't know (if other teams are interested) to be honest, but if they are then I wouldn't be surprised," said Leicester's owner.
"As long as he's fit, and my manager gives me a green light on that, and he's ready to go and he has the desire to come back to England, age is relative as long as the player is fit and can last a full game. I don't see any problems with that though."
Meanwhile, Mandaric insists that Pearson will not be leaving the club to take over at Championship promotion rivals Middlesbrough.
Pearson played for Boro between 1994 and 1998 and captained them to two promotions and three domestic cup finals during his time on Teesside.
However, Mandaric, who allowed Gary Megson to leave the Walkers Stadium to take charge at Bolton two years ago, is confident that Pearson will remain at the club.
"I wouldn't put money on him going there, it's not going to happen. He is our man here, he's happy with us and we are happy with him," said Mandaric.
"On the other hand, I would be surprised and unhappy if people didn't look at Nigel because he is a good manager. But he is as solid as a rock as far as that goes and he is staying with Leicester City."