Rodriguez, who scored 22 goals for the Baggies last season, is on the radar of several sides.
Bournemouth are interested in Rodriguez,and could turn to him as an extra body up front.
Rodriguez worked with Eddie Howe at Burnley, and enjoyed his best spell for the Clarets under the current Bournemouth manager.
The 29-year-old could leave Albion this summer after spending a season in the Championship, playing 47 out of 48 league games last season – including the two play-off matches.
His hometown club Burnley, who made a £16m bid to re-sign their academy product last summer, could also return for the versatile forward although they're yet to make any approach.
Rodriguez's work ethic and appetite has impressed those behind the scenes at Albion throughout the campaign.
It was his first in the second tier in seven seasons, and even though he has two years left on his contract he is one of a number of players who could exit.
Albion wanted around £20m for Rodriguez last summer, but he could be available for less this time around.
Technical director Luke Dowling admitted after the season finished: “Next season it will be slightly different. We’ll take the right risks in player recruitment that will benefit the club.
“If it means two, three, four, or five players move on, then they move on. We have to back ourselves to replace them.
“We had a group of players who we believe can play in the Premier League. When I joined in September, they as a group were willing to give it this one season.
“A lot of them, with the age that they’re at, will be thinking ‘if I don’t go to the Premier League now then I could spend the rest of my career in the Championship’.
“We were a penalty away from getting to Wembley but we have fallen short.”