The West Ham United winger enjoyed a fruitful loan spell at Villa during the 2017-18 campaign, scoring seven goals and assisting a further 14 in 43 Championship outings.
Since returning to the London Stadium, Snodgrass has featured in several of West Ham's pre-season games, seemingly putting him in contention to feature in the Premier League.
According to the Birmingham Mail, however, Steve Bruce remains eager to re-sign the 30-year-old on loan to bolster their promotion hopes.
The report also claims that new Hammers boss Manuel Pellegrini may be willing to let Snodgrass leave at the end of August, with EFL clubs still able to complete domestic loans up until August 31.
Snodgrass has also been linked with a move to Premier League side Cardiff City over the summer.