Jurgen Klopp attempted to bring the Real Madrid attacker to Anfield ahead of the current campaign, but the La Liga giants refused to countenance a sale.
However, Asensio has endured a difficult season at the Santiago Bernabeu, scoring just one goal in 28 league apperances and completing 90 minutes on just nine occasions.
And with Madrid plotting a major squad overhaul this summer, Liverpool believe they could succeed in prising Asensio away from the Spanish capital.
Florentino Perez is keen to keep hold of the 23-year-old, but the Reds are hopeful that a concrete offer could persuade the Madrid president to change his position.
Liverpool will play in the Champions League final for the second year in a row following Tuesday’s astonishing 4-0 victory over Barcelona.
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