The Brazilian international is expected to be surplus to requirements at the Santiago Bernabeu, with Jose Mourinho not lacking in midfield alternatives.
The Real Madrid is understood to be keen on letting Kaka go, but only on a permanent basis, for which Arsenal would have to stump up around £20m.
The big signing of the creative midfielder would be uncharacteristic of the Arsenal manager, especially given that Kaka is 29 years-old and presents little sell-on value.
Kaka is currently on £153,000-a-week and may have to take a cut in wages if he were to complete a move to the Emirates. However, the Gunners do have funds available after selling Fabregas and Nasri to Barcelona and Manchester City, respectively.