Parkinson has been out of work since quitting Bolton, who are now Sunderland's League One rivals, in August.
It is claimed the Mackems want a manager with third-tier experience and the 51-year-old has that in abundance, having previously led Bolton and Colchester United into to the Championship.
But he is not the only boss under consideration, with Gareth Ainsworth and Paul Cook, of Wycombe Wanderers and Wigan Athletic respectively, also being looked at, along with the out-of-work duo, Daniel Stendel and Nigel Pearson, according to The Sun.
Interestingly, the newspaper adds that both Ainsworth - whom separate reports claim Sunderland have been granted permission to hold talks - and Cook are keen on the job vacated by Jack Ross earlier this week.
Parkinson has also had spells at Hull City, Charlton Athletic and Bradford City - with whom he won promotion to League One.
Were Sunderland to appoint the Lancastrian, his first meeting with Bolton would be at the Stadium of Light on Boxing Day.
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