'Sky Sports News' reports that the deal is very near to completion, with an announcement possible as early as Monday.
The fee is believed to be £15m, £12m of which will be paid up-front and a further £3m in potential add-ons.
'Baggies' boss Tony Pulis is a long-term admirer of the 27-year-old England international and it seems like he will finally get the chance to work with the player.
Rodriguez has found regular game time hard to come by in the last couple of seasons and it is understood that this would have continued under new boss Mauricio Pellegrino, with Manolo Gabbiadini, Shane Long, Charlie Austin and Sam Gallagher making up the striking department at St Mary's.