The England under-19 international completed a medical with the FA Cup holders on Friday after making 25 senior appearances for the Saints last season.
It is believed that Wenger sees the Englishman as cover in the right-back position and as a future international star. The large transfer outlay dedicated to the deal, however, may cause derision from some areas with supporters calling for high-profile signings at the Emirates.
Reports also suggest that Arsenal have sounded out Chambers' teammate Morgan Schneiderlin.
The 24-year-old midfielder, who represented France at the World Cup, has been a long-term target for the Gunners.
Should the Chambers deal come into fruition, the defender would become Arsenal's third acquisition of the summer after Barcelona forward Alexis Sanchez and Newcastle right-back Mathieu Debuchy both signed on.
Southampton, who have already lost Rickie Lambert, Adam Lallana and Luke Shaw this summer, also look set to sell Dejan Lovren as the defender nears a £20m move to Liverpool.
