Robins boss Andy King was keen to the former Nottingham forest man to a deal at the County Ground with the Wiltshire side aiming to improve on last season's form.
King has also offered Brazilian midfielder Stefani Miglioranzi a similar one-year deal with both he and Edds having settled in well at the club.
King told The Evening Advertiser: "Gareth (Edds) and Stef (Miglioranzi) have clearly earned the contracts they are now thinking over.
"They have both improved my squad and they have done well in the matches we have played so far.
"Gareth's commitment to the cause was evident to me when he said he would only be paid on an appearance basis.
"When a boy says he will effectively play for nothing, that tells you a lot about his character.
"Stef has been superb on the pitch and I cannot fault his commitment. I hope he agrees to the deal on offer.
"Quite simply, both players have shown a loyalty to this football club and I think it is right that we demonstrate our loyalty to them."
Meanwhile, veteran Neil Ruddock has stated he will not be forced out of the County Ground despite the club's desire to get rid of the former England man's wage bill.
The 34-year-old is determined not be forced out by The Robins and aims to prove his worth to Andy King's side.
Ruddock said: "I take the attitude that it's a case of sticks and stones may break my bones but names will never hurt me.
"Actually, it is upsetting to hear people saying unkind things, but I've put up with all sorts of stick throughout my career.
"All I can do is try and show people that I am committed to Swindon Town by my performances out on the pitch."
In other news, Robins trainee Graham Walton has dismissed from the club's youth system on the grounds of misconduct.
