The Daily Mail is reporting the Reds are happy to let Vigouroux turn his season-long loan with the League One outfit into something more long term.
The decision comes after Liverpool re-signed first choice shot-stopper Simon Mignolet while also recalling Danny Ward from his loan spell at Aberdeen and signing Polish youth goalkeeper Kamil Grabara.
Swindon chairman Lee Power said his club were eager to secure the services of Vigouroux, should the Reds be willing to complete the permanent transfer.
"There has been a change there and there has been a change here with us, but we really like Lawrence," Power told BBC Wiltshire.
"He is a bit of a wildcard and he is one of those. We discussed internally that you could sell Lawrence one day for a fortune. He reminds me a little bit of Bruce Grobbelaar and you don't know what you are going to get with him.
"It might not be great sometimes but other times he takes your breath away with things that he does, but he is someone who we have spoken to Liverpool before about.
"They said they wouldn't sell, but we are going to try and have another crack at him in the window. We are trying to crowbar the door open at the moment as we speak."