The former Red Devils boss was sworn in at the Stadium of Light on Saturday.
The Scot is aware that a new centre-forward and right-back are his main priorities ahead of the Mackems' Premier League opener at Manchester City in 14 days time. However, other positions are seemingly still under review.
Moyes has been linked with reunions for Manchester United pair Adnan Januzaj (Mail) and Marouane Fellaini (Sun) since taking over, however, the report adds he is yet to make any enquiries for either midfielder.
Januzaj made his debut under Moyes at United and marked his first league start, ironically, with two goals against Sunderland, while Fellaini was signed by the 53-year-old at both Everton and Old Trafford.
The Sunderland Echo reported earlier this month that both a number 10 and central midfielder were being sought by Allardyce, and the pair would fit the bill should Moyes share the same view.
The Black Cats today sanctioned Adam Matthews' return to Bristol City on loan, which suggests the club are confident of landing a replacement in time, with Billy Jones their only remaining right-back.
Moyes is also short of options up top, with Fabio Borini his only out-and-out striker with Jermain Defoe sidelined, though he should return ahead of the club's trip to the Etihad Stadium.