GWS have made clear their interest in the 30-year-old, who is contracted to the Bombers for next season.
There has been growing speculation that Stringer will go to the Giants on a two-year contract.
But on Thursday, Essendon list manager Matt Rosa said they have received "no formal position" from Stringer's camp on wanting a trade.
"As we're aware, the Giants have shown some interest," Rosa told AFL trade radio.
"He's played some really good football for us this year and the fact he's locked into a one-year deal is where it sits."
Rosa added Essendon would need "adequate compensation" for Stringer, who played 23 games this season for a career total of 212.
Stringer featured in the Western Bulldogs' 2016 premiership team before he joined the Bombers at the end of the following season.
Also on Thursday, GWS youngster Conor Stone rebuffed interest from Essendon and re-signed until the end of 2026.
Conor Stone (l) has signed a new deal keeping him at GWS for the next two seasons. (Dean Lewins/AAP PHOTOS)
Essendon had met with Stone, who has played 13 senior games, and he could have been part of a trade deal for Stringer.
"We clearly had some interest in Conor ... we probably weren't surprised he decided to stick with the Giants, given (their) list changes. He can see real opportunity there," Rosa said.
The trade period ends on Wednesday and the likelihood is that the major deals involving Bailey Smith and Don Houston will not happen until next week.
Smith wants to leave the Bulldogs for Geelong and Port Adelaide's Houston is keen to return to Victoria, with Collingwood for now his most likely destination.
Smith's teammate Jackson Macrae also wants a trade to St Kilda, who re-signed Dougal Howard and Zak Jones on one-year deals.
Also on Thursday, Richmond secured key position player Noah Balta for the next eight years.
Noah stopping him. — Richmond FC � (@Richmond_FC) Enjoy the best of Balta, as he extends his contract until 2032 � pic.twitter.com/QAZEIjoAOCOctober 10, 2024
With the Tasmanian expansion team looming, Balta is the latest player to commit on a long-term contract.
The 24-year-old was already signed for next season and his new seven-year extension ties him to Punt Rd until the end of 2032.
Balta has played 104 games for the Tigers, including the 2020 premiership in his second season.
"I think I am a loyal person, and I want to repay the favour that players like David Astbury, Alex Rance, and Dylan Grimes showed me when I was younger," he said.
"They pushed me to get better, and now I want to be the one helping to drive the next generation. I could not see myself playing anywhere else."
Richmond lost Jack Graham to West Coast through free agency this week, while Liam Baker (Eagles), Shai Bolton (Fremantle) and Daniel Rioli (Gold Coast) want trades.
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