The report claims that Southampton are interested in signing Gaitan during the January transfer window, alongside two other Premier League sides.
Gaitan looks set to leave Madrid after the Spanish club signed Vitolo and Diego Costa this week.
There were reports that Gaitan is available for £13.3m, although Southampton appear to be going down the loan route.
Southampton received a world-record fee for a defender after the sold Van Dijk to Liverpool for £75m earlier this month.
The Saints are looking to strengthen their attacking options this month, having failed to win any of their last nine matches in the Premier League.
The club currently sit one place above the relegation zone, with only goal difference keeping them away from the drop.