The Mirror says Snodgrass impressed on loan last season with seven goals and 14 assists in 43 Championship games for Villa.
He was a key part of Bruce's side who came fourth and missed out on promotion in the play-off final.
Scottish international Snodgrass,30, has since returned to Premier League West Ham.
He even starred in the Hammers' 3-1 friendly success over Villa last month at Walsall.
But Snodgrass is expected to be allowed to leave West Ham later this month on loan after new boss Manuel Pellegrini made eight new signings.