Southampton are in the process of signing striker Ross Stewart from Sunderland and wanted Roberts too.
They put a package of £5m on the table for Roberts, but Sunderland believe it undervalues the winger and have rejected it.
Southampton will not be going back with an improved offer for Roberts and have pulled out of the chase for the former Celtic man.
Sunderland are keen for Roberts to stay and it appears he will be remaining at the Stadium of Light.
Roberts has featured in three of Sunderland’s four Championship games so far this season.
He has entered the final year of his deal at the Stadium of Light.
It remains to be seen if Southampton might try again for Roberts when the January transfer window swings open for business.
