Sunderland have had a scout at five matches this season but could find competition from Middlesbrough and a host of Championship clubs as well, who would have been impressed with his performances at right back and in the centre of defence.
27-year-old Woodards was a product of the Chelsea youth team but failed to break in to the team at Stamford Bridge and ended up at St James's Park after his release from Chelsea.
Woodards signed a three year deal in January after the Railwaymen beat Yeovil Town to his signature, as part of the deal Exeter had a 20% sell on fee inserted in to the contract.