The Tykes are seeking a replacement for Paul Heckingbottom, after the 40-year-old's departure on Wednesday.
And it is claimed that Moore, who like Heckingbottom is a former Barnsley player, is among those on the Reds' list to succeed him.
The Sun reports, however, that it would take a lot for Moore to leave West Brom, having only been promoted to the role of Baggies first team coach in December.
Nevertheless, it adds that an offer from Barnsley would leave the 43-year-old facing a dilemma, especially given his connection with the Championship strugglers.
Working down the list?
Barnsley have been linked with a string of names in the wake of Heckingbottom's departure, which appears to have caught his former employers by surprise.
HITC Sport understands they have already been knocked back by Danny Cowley, the Lincoln City manager, while linked pair Grant McCann and Karl Robinson, of Peterborough United and Charlton Athletic respectively, may also prove difficult to prise away.
It is perhaps likely that Barnsley would have more success with someone like Moore at West Brom or David Unsworth at Everton, neither of whom have held senior managerial roles previously.
Could Moore leave West Bromwich Albion for Barnsley?