The Daily Star says talks between the two parties went well on Wednesday, with Keane being “receptive" to the idea of a return.
The Irishman is understood to have asked for time as he mulls over his options.
He has enjoyed a brilliant career as a pundit since his last management position at Ipswich Town in 2011.
Keane will now have to decide whether to give up his lucrative contracts with Sky Sports and ITV, having first heard about Sunderland's interest on Monday.
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