The Real Madrid forward has been strongly linked with a move away from the Bernabeu after falling down the pecking order under Zinedine Zidane.
The Frenchman even dropped Rodriguez from his his matchday squad in Madrid's Champions League win over Juventus on Saturday.
A lack of game time has led to suggestions he will be playing his football elsewhere next term with Manchester United mooted as his next destination.
But Spanish publication Don Balon claim Liverpool are ready to battle their rivals for his signature.
Liverpool are reportedly willing to offer £52m for Real Madrid's James Rodriguez James Rodriguez transfer odds Mon, May 15, 2017
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Paris Saint-Germain, AC Milan and Bayern Munich are also chasing the attacker.
Madrid chief Florentino Perez is refusing to lower his £52m valuation of the Colombian that has proved to be stumbling block in negotiations with United.
But Don Balon say Liverpool are willing to meet his price-tag.
James Rodriguez is thought to be keen to work with Jose Mourinho at Manchester United
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However, it seems convincing the 25-year-old to choose a move to Anfield over Old Trafford could prove difficult.
Rodriguez is said to favour a switch to the Red Devils and is keen to work under Jose Mourinho.
The report adds if Rodriguez's move to United falls through his second preference would be to join Carlo Ancelotti at Bayern Munich.