Saints are signing Ross Stewart from Sunderland to bolster their attack and also want to bring in winger Roberts.
Losing Roberts would be a blow for Sunderland, but they are prepared to discuss selling him.
Southampton got in touch on Thursday to offer Sunderland a package of £5m to take Roberts to the south coast before the window closes.
Talks between the two clubs are continuing and it is suggested that it is possible an agreement could be reached.
Sunderland are looking to bring players in and would likely want to replace Roberts if he does go to Southampton.
Roberts, 26, is into the final year of his contract at the Stadium of Light.
The former Celtic man has already featured on three occasions in the Championship for Sunderland so far this season.
