Benitez's interest in Keane is an unexpected development but, with Crouch likely to go and Portsmouth still his most likely destination and Andriy Voronin struggling to make an impact, he is seen as an ideal partner for Spain's Euro 2008 hero Fernando Torres.
Keane is under contract with Spurs until 2012, having signed a new five-year contract in May 2007.
He was an influential figure last season, scoring 23 goals in a campaign that was capped by the Carling Cup going to White Hart Lane - and Liverpool's move will receive no encouragement.
But reports suggest Keane may be tempted to make the switch by the lure of playing Champions League football at Liverpool, the club he supported as a boy.
He joined Tottenham from Leeds for £7m in August 2002 and has also played for Wolves, Coventry and Inter Milan
Liverpool, meanwhile, may have to wait until European champion Xabi Alonso returns from holiday before they can sort out the Spanish midfielder's future.
Alonso is on a three-week break after being part of Spain's successful Euro 2008 squad, leaving a possible £16m move to Juventus in limbo and boss Benitez waiting to discover if he will get a fresh injection of transfer funds.
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