Taylor is also wanted by Scottish giants Celtic and they remain in talks.
The Daily Record report: "Angelo Alessio has confirmed that Kilmarnock rejected a bid from Sunderland and they are still in talks with Celtic.
"The left back is a target for Neil Lennon as he looks to replace Kieran Tierney."
Kilmarnock want £3m for their key player. Sunderland have until 5pm Monday to sign players.