The West Bromwich Albion star is available through a release clause in his contract, which was activated when they did not achieve promotion to the Premier League.
However, the clause states the fee will rise to £10 million after 12 months, which means it is not as low as the initial £5 million clubs expected.
According to the Burnley Express, the deal must be done by mid-July for the clause to remain valid.
Norwich City are among five or six clubs vying for Rodriguez's signature, per previous reports in the British media.
The veteran striker managed more than 20 goals for West Brom in the Championship last season.