The 26-year-old played a key role in Chile’s recent Copa America triumph and has 40 caps for his country, scoring six goals.
Aranguiz would help strengthen Leicester’s midfield following the departure of Esteban Cambiasso.
If completed, the deal will see Aranguiz become the club’s record signing, with his transfer exceeding that of Andrej Kramaric, who joined the club in January for a fee in excess of £9m.
Leicester have moved to further strengthen their midfield by agreeing a fee of £3m for experienced Napoli midfielder Gokhan Inler.
The 31-year-old has amassed 83 caps for Switzerland and has made 236 Serie A appearances during eight seasons with Udinese and Napoli.
Leicester manager Claudio Ranieri revealed earlier this week he was close to securing a replacement for Cambiasso and now it appears he is targeting two.
“We are still looking. We are very close with a good player,” Ranieri said following his side’s 2-1 pre-season victory over Burton on Tuesday.
When asked about the challenge of replacing Cambiasso, Ranieri added: “Yes, of course, because Esteban is a good player, a leader and it is difficult to replace him. But we are looking for him.
“We are very close (with a replacement). But I wait until the end of the week.”