Moyes has worked with Naismith in the past, bringing him to Everton in 2012 from Rangers.
The 29-year-old left Goodison Park for £8.5 million, but he was unable to help guide Norwich towards safety with the Canaries ending the season in 19th position.
Sunderland are in desperate need of a forward, with Jermain Defoe the only out-and-out first-team striker in the squad right now.
Despite not having spent even a year at Norwich, Naismith could be tempted by a reunion with Moyes in the Premier League.
Norwich got their season off to the best possible start on Saturday away to Blackburn as they begin their quest for promotion.
Having gone through the ecstasy of winning 4-1 away from home, it could turn to agony for Norwich fans if they were to lose Naismith. Fans of the Canaries have taken to Twitter to express their displeasure at the rumours.
I must admit, if a club had come in with £8 million for Naismith over the summer I'd have been delighted. Now I'm much more hesitant. #ncfc— Edward Jacobs (@Edward_Jacobs_) August 10, 2016
I'd be annoyed if we sold Naismith. I really think he will play a big part this year and have an impact on the division.— Craig Daniels (@craigdaniels82) August 10, 2016
Sunderland can keep their money. I would not sell Stevie Naismith #ncfc— Tim Scales (@tim_ncfc) August 9, 2016
Very quite on the striker front and these rumours of Naismith to Sunderland want to f**k off! #ncfc.— Jamie Burgum (@jamieburgum) August 10, 2016
While Norwich fans do not seem keen on the idea of losing Naismith, Sunderland fans are not too happy with the link themselves.
I hope the reports got mixed up & we're offering Norwich 8 packets of Wotsit's instead of £8 million quid for Naismith #SAFC— John Raine (@ChoccyRaine) August 9, 2016
If Naismith isn't good enough for Norwich then why should he be good enough for us?— Jack (@jacklemore123) August 9, 2016
Steven Naismith. Come on man Moysie— Ian Clementson (@iclementson) August 10, 2016