Cork, 24, has 18 months remaining on his contract with the south coast side and has become a regular starter in recent weeks, following the injury obtained by Victor Wanyama against Aston Villa in early December.
A series of effective and impressive performances, for which Cork was highly-regarded for after forming a formidable partnership alongside Morgan Schneiderlin, has led to interest from other clubs.
Swansea contacted Southampton to make a bid worth in the region of £3m for the English midfielder, but they failed in their attempts to lure Cork away from the south coast.
The South Wales side saw their offer fall foul to an outright snub from Southampton's hierarchy, who have absolutely no intention of selling Jack Cork, or any other first team player.
Southampton chief Mauricio Pochettino warned suitors that no player that he wanted to keep would leave the club in January, earlier this month.
Today, he reiterated, as quoted by Daily Mail: 'Jack Cork is a very important player for us and there is no possibility of him leaving the club.'
One potential outgoing from St. Mary's Stadium is Dani Osvaldo, whose position is untenable due to a training ground incident with Jose Fonte last week.
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