Leicester reportedly made an initial bid of £25m last week, but it was swiftly rejected by Juventus.
According to Corriere dello Sport, the Foxes latest bid is €40m (£34m) which is a significant increase on the previous offer.
The problem for Juventus is that Leicester will reportedly only pay €30m upfront and €10m in add-ons.
The Old Lady are reluctant to let who they now regard as a prized asset leave without receiving plenty of money up front.
Demiral has gained recognition in the past year following impressive performances, especially with the Turkish national team.
The 21-year-old started the season on the bench, but has steadily established himself in Turin.
And now Demiral has gained the trust of Juventus manager Mauricio Sarri, who reportedly rates him above their expensive summer signing Mathjis De Ligt.
In the heart of Turkey's defence, Demiral plays alongside Leicester's Caglar Soyuncu and could extend the partnership to club level if a deal happens.
It's widely reported that Leicester's Premier League rivals Arsenal are also considering a move for Demiral.
The Gunners are seeking to plug a hole in their leaky defence and are believed to have enquired about Demiral last summer.
Now that Calum Chambers has been ruled out for up to nine months with a cruciate ligament injury, reinforcing central defence is even more of a priority.
Juventus are yet to reply to Leicester's latest offer, but it's understood they currently value Demiral too much to let him go mid-season.
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