Snodgrass is understood to be surplus to requirements at West Ham, despite only joining the Hammers in January in a £10million deal from Hull.
And Bruce, who previously signed Snodgrass in 2014 during his time in charge of Hull, admits he would be open to a reunion with the Scotland international at Villa Park.
Aston Villa will have to compete with Championship rivals Birmingham City and Sunderland for the 29-year-old’s signature.
Villa have also been linked with a move for Arsenal midfielder Jack Wilshere, but Bruce doubts the England star is willing to drop down a division.
“Snoddy is somebody I’ve known for a long time – I bought him for Hull – so that’s something we’re looking into and exploring,” he told talkSPORT.
“The Jack Wilshere one – hey, who wouldn’t want Jack Wilshere? But I don’t know whether we’re capable of landing him and let’s not forget he’s only 25.
“People have said that I’ve got millions and millions to spend, where it’s been the complete opposite.
“With financial fair play, unfortunately we haven’t got millions and millions to spend so I’ve got to be a little bit prudent, see what we can do and try and balance the books.
“In that respect it’s been a very difficult summer, but an enjoyable one. I think the vast majority of supporters understand that.
“If they think I’m going to breeze in just before last Christmas and turn it around within six months, that ain’t gonna happen.
“But thankfully we’re starting to have the makings of a very decent squad.
“So I’m very confident of where we sit at the minute and very confident we’ll make a fist of it.”