The Saints are ready to spend the money they have received from the sales of key players this summer, and Armero is being targeted as an ideal replacement for young left-back Shaw, who has joined Manchester United.
Armero played in every game for Colombia at the World Cup before they were knocked out of the competition by Brazil last night, and Southampton are said to be ready to swoop.
The 27-year-old is registered with Udinese after the Italian club bought him outright just last month, as he was co-owned by Napoli beforehand.
He spent the end of last season on loan at West Ham United, making five appearances in the Premier League under Sam Allardyce, and is reported to be desperate to get back to the Premier League on a permanent basis, now that his World Cup campaign is over.
The Colombian has one year remaining on his current contract at Udinese, who could be willing to cash in on the pacey left-back this summer.
Arsenal are also believed to be interested in his services, but new Saints boss Ronald Koeman has money to spend and can offer Armero first-team football on a regular basis if he moves to St Mary's.
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Armero scored Colombia's first World Cup goal with his only shot of the tournament in the opening game against Greece and was a constant threat down the left side.
He would be an ideal replacement for Shaw because of his attacking mentality and willingness to create chances for others around him.