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He is set to arrive in England later today as Leicester look to put him through medical and the other formalities of a transfer.
Akgun would have to be registered before noon today if Leicester want him involved against Rotherham on Saturday.
Leicester have been chasing the winger for several weeks and they have a deal in place to sign him this summer.
The Foxes have been pushing to complete the deal but Galatasaray wanted him for their Champions League qualifier against Molde.
He provided an assist in their 3-2 win at Molde on this week and is now set to join the Championship side.
According to Turkish outlet Sports Digitale, the winger has left Istanbul and is on his way to Leicester this summer.
Leicester have agreed on a loan deal with a mandatory purchase clause worth €10m with Galatasaray.
Akgun has also worked out personal terms on a contract and has a draft agreement for a five-year deal.