The Celtic midfielder is believed to be ready to bring a long-running transfer saga to a close by agreeing a switch to the Saints.
It is understood that he will travel to the south coast on Wednesday evening, where the final details of a transfer will be thrashed out.
Southampton will want to get a deal over the line as quickly as possible as Wanyama has been interesting a number of their Premier League rivals.
Cardiff City were in the market for the Kenya international at one stage, but appear set to be left empty-handed.
Snapping up Wanyama will be some signal of intent by Southampton, with the 22-year-old having seen his profile rocket since arriving in Scotland back in July 2011.
While Celtic will be disappointed to see him leave Parkhead, they are set to make a considerable profit on a player signed from Belgian outfit Beerschot AC for just £900,000.