Albion boss Tony Pulis finally looks set to land the 27-year-old a year after making his first bid. Pulis looked set to sign the former Burnley man on loan last summer, only for the Saints to pull the plug at the 11th hour.
West Brom failed in another bid to sign him on loan in January but Pulis now looks set to achieve third time lucky.
Saints head of football development Les Reed admitted Rodriguez could leave but insisted it will not be the start of a mass clearout.
Reed, who also claimed defender Virgil van Dijk would not be sold, said: “There may be players that go on the basis that it’s the right thing for them and for the club at a given time. But we don’t expect that to be wholesale.
“This is simply doing sound business in the transfer window.”
Pulis also wants a centre-back with Middlesbrough’s Ben Gibson one of his top targets.
But he is likely to face stiff competition for the defender, who looks set to leave the Riverside Stadium this summer following Boro’s relegation.
West Brom are also targeting midfield reinforcements after losing captain Darren Fletcher to Stoke on a free transfer.
