The Paraguayan was linked with both Premier League clubs heavily over the course of the closing stages of the transfer window, with the Hammers in particular interested in his signature.
Although the 30-year-old was believed to be ready to leave the Portuguese giants, but it has now emerged that he opted out of moves to join Sam Allardyce’s side or the Saints.
The experienced South American has been with the Lisbon side since 2007, where he has forged a reputation as one of the best goal getters in European football.
This alerted ‘Big Sam’, who was desperate to land a striker to fill in for the injured Andy Carroll.
Southampton chief Mauricio Pochettino was also keen, despite having signed Pablo Osvaldo from Roma.
Cardozo is believed to be happy to be staying put, but could be the subject of interest again in January, with Turkish outfit Fenerbahce and an unnamed side from Qatar monitoring his situation.
It is unclear as to whether West Ham and Southampton will revive their interest, with much depending on their respective starts to the season.
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