The striker is a long-term target for Baggies boss Tony Pulis and he can leave Saints this summer.
Press Association Sport understands Albion have resurrected a move with £12million likely to be enough to sign the England international.
Rodriguez has almost left St Mary's in the last two windows with Watford, Hull and Sunderland all keen on him in January but Hull and Sunderland's relegation has ruled them out of the running.
Had the 27-year-old left Southampton in January West Brom were frontrunners.
He has wanted to play regularly at Southampton but has struggled for game time since suffering a serious knee injury in 2014.
Rodriguez made 34 appearances for Saints last season but only nine were Premier League starts and he scored six times.
Albion are looking for new firepower after relying on Salomon Rondon and Hal Robson-Kanu as their only two strikers for the majority of last season.
Saido Berahino was sold to Stoke in January but missed the first half of the season having served a drugs ban and being ruled unfit by the club.
Source : PA
Source: PA