The Reds meanwhile are in desperate need of goals, having scored just once in their last four matches on the road.
And Murphy thinks Rodriguez would be an ideal addition to the squad if Jurgen Klopp can lure him to Anfield.
"Pace is a commodity feared by defenders and yearned for by supporters for its ability to get people out of their seats," Murphy told talkSPORT.
"At the moment, Liverpool’s attack looks blunted while Daniel Sturridge’s injury problems continue to frustrate. Christian Benteke is undeniably gifted but, as illustrated by a turgid display at West Ham, his immobility doesn’t seem to fit in with Liverpool’s mantra.
Liverpool club fact file
"A highly ambitious target would be Pierre-Emerick Aubameyang (left) at Borussia Dortmund – the highest scorer in Europe’s top five leagues so far this campaign – but it would take over £40m and some serious seduction methods to lure him to Anfield.
"A more realistic alternative is Jese Rodriguez (right) at Real Madrid, who has grown tired of a lack of game time in the Spanish capital. Jese is young, quick and well worth a gamble in January given his current discontent."
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