The Mirror reports that the Clarets have tabled a £25m bid to sign both players as West Brom could pay the price for relegation by losing two of their best players.
Burnley finished 7th in the Premier League to qualify for the Europa League, their first experience of European football since 1966/67. If they make it through to the group stages, it will place greater stress on the squad and they will need more strength in depth.
For Rodriguez, it would be a return to the club where he made his name, scoring 31 league goals for the Clarets between 2008 and 2012.
He then moved to Southampton, where he was on the periphery of the England squad and made his debut before a serious cruciate ligament injury ruled him out for over a year.
He does have experience of playing and scoring in the Europa League though, having done so for the Saints in 2015 and 2016.
He has never been the same since the injury but he was West Brom's joint-top scorer last season, although he couldn't prevent them from finishing bottom of the league.
The man who shared the top scorer award with him was Salomon Rondon, but the Venezuelan could also leave the Hawthorns if Newcastle meet his £16.5m release clause.
Having already lost Jonny Evans to Leicester for just £3.5m, it is shaping up to be a tough summer for Darren Moore and a real baptism of fire in his first managerial job.