Norwich's Steven Naismith in action with Watford's Ikechi Anya
The Black Cats have made just one signing so far this summer as defender Papy Djilobodji arrived from Chelsea last Friday, strengthening Moyes' options at the back.
Norwich's Steven Naismith and Newcastle's Cheick Tiote
However, Moyes needs to find some support for leading goalscorer Jermain Defoe, and he may turn to a familiar face to help him on the goalscoring front.
The Daily Star report that Sunderland scouts were in attendance as Norwich City beat Blackburn Rovers 4-1 on Saturday afternoon, with forward Steven Naismith believed to be their target.
Tim Howard in action with Norwich's Steven Naismith
Naismith, 29, was signed by Moyes at Everton back in 2012 after he decided to leave Rangers following their financial collapse, and he scored four goals in 35 games in his only season under Moyes.
The Scotland international hit 25 goals in 124 games for Everton before moving on to Norwich in January, and he scored once in 13 appearances for the Canaries as they failed to avoid relegation.
Steven Naismith of Norwich City celebrates scoring
Now in the Championship, Naismith started against Blackburn on Saturday, and will have impressed Sunderland scouts by scoring the fourth goal of the game for Alex Neil's side.
It's now believed that Moyes will look to mount an £8m move to reunite with Naismith, which would add goals and creativity to the Sunderland side behind the potent Defoe.
Would you like to see Sunderland sign Naismith for £8m?