The Daily Mail reports that Southampton scouts watched the 21-year-old during last week’s game against Motherwell.
Wanyama has also been strongly linked with the likes of Liverpool and Manchester United in recent months.
The Kenyan midfielder is valued at around £10million by Scottish side Celtic.
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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.