The Black Cats are in need of cover at left back and have been trying to broker a deal with Fiorentina for Marcos Alonso, who spent the second half of last season on loan at the Stadium of Light.
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But they have now focused their attentions on Van Aanholt, who impressed as part of Chelsea's loan contingent at Eredivisie side Vitesse Arnhem last season.
The 23-year-old moved to Stamford Bridge as a teenager in 2007 but has failed to establish himself in the first-team squad, and has spent time on loan at five different clubs in as many years.
This summer saw Ashley Cole depart after nine seasons at Chelsea but Jose Mourinho immediately moved to sign Filipe Luis as his replacement, limiting Van Aanholt's chances of securing a starting spot.
Former Blues star and Sunderland boss Gus Poyet is also believed to be keen to bring Kurt Zouma and Patrick Bamford to the Stadium of Light on loan, after both produced eye-catching preseason performances.
Poyet has already signed goalkeeper Costel Pantilimon from Manchester City, midfielder Jordi Gomez from Wigan Athletic and defender Billy Jones from West Brom this summer.