Foxes boss Enzo Maresca wants to snap up the Turkey international and the club have been working hard to deliver.
Progress has now been made on a switch for Akgun to England, with Leicester pushing a step further on the deal, according to Turkish journalist Salim Manav.
Unless there is a hiccup, Akgun is expected to complete a loan move to Leicester in the coming days and the Foxes will have an option to buy included in the agreement.
Akgun, 23, helped Galatasaray to win the Turkish Super Lig title last season.
He made 25 appearances in the Super Lig over the course of the last campaign, scoring once and providing four assists.
Akgun has never played his football outside Turkey and will be looking to quickly get to grips with the Championship.
The attacker has been capped by Turkey at international level on five occasions.
