Southampton have had a club-record £25million bid for Spartak Moscow forward Quincy Promes turned down.
Saints boss Mauricio Pellegrino has been chasing the Holland international throughout this transfer window.
But the Russian side have told them they will have to break their transfer record for a second time this month if they want to sign the winger.
Their previous club record outlay was £18.1m for Mario Lemina last August but they smashed that with the £19.2m signing of Monaco forward Guido Carrillo last week.