Boss Mauricio Pellegrino wants to sign his fellow Argentine on loan to bolster Saints’ hopes of avoiding relegation.
Former Benfica star Gaitan, 29, is looking to quit the Spanish capital to save his World Cup place after falling out of favour this season.
Pellegrino will face competition for former Manchester United target Gaitan from Crystal Palace and Everton.
Fans’ support for their manager appears to be wearing thin, though, despite earning victory over Fulham to advance to the fourth round of the FA Cup.
It was the club’s first win in 10 matches and captain Steve Davis has called for the supporters to get behind Pellegrino.
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“Stuff like that doesn’t really help,” Davis said.
“We need the fans more than ever at this point in time – whenever you’re struggling.
“To be fair, I thought they were in good voice today as well, and came here in good numbers.
“It’s important that we all stick together. We’re not in a position that any of us want to be in, fans, players and staff alike.
“The most important thing is we’re all fighting together to move forward in the right direction.”
Saints, meanwhile, remain confident of bringing back Arsenal outcast Theo Walcott to St Mary’s, despite Arsene Wenger insisting he wants to keep the player.