Reports suggest the Gunners have already registered their interest in the England forward, and were considering a move before he suffered the knee injury that wrecked his World Cup dream.
The club will now monitor his recovery progress, and remain interested in signing him at some point in the future.
Liverpool and Tottenham are both said to be interested in the 24-year-old, who was ruled out of action for six months following the injury sustained in the 4-1 defeat to Manchester City on Saturday.
However the forward is confident of making a full recovery, and may even return a month ahead of schedule.
Rodriguez emerged as a target for the Saints’ Premier League rivals after firing 17 goals this season, and putting himself firmly in contention for Roy Hodgson’s World Cup squad.
However, Southampton are keen to allay any doubts over his future, by signing the versatile forward up to a new five-year-contract at St Mary’s.
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