The Uruguayan international, understood to want a move away from the Premier League after being involved in several controversial incidents while at Anfield, is hopeful the English giants will let him leave at an affordable price.
This would likely be around €25m plus one of Madrid’s youngsters moving in the opposite direction, with Reds’ boss Brendan Rodgers thought to be keen on either Jese or Alvaro Morata, although the Spaniards would retain an option to buy both strikers back.
Los Blancos plan to offer 26-year-old Suarez a four-year deal at the Santiago Bernabeu but they could face late competition from Serie A, with Juventus, Inter and Napoli all believed to be interested in the player.
Nonetheless, his in-laws live in Castelldefels, 20 kilometres to the south-west of Barcelona and he place he sometimes goes to escape the English winter, so he would be particularly receptive to a move to La Liga.
Suarez joined Liverpool from Ajax before the start of the 2010-11 campaign and was second-highest scorer in last season’s Premier League with 23, three behind Manchester United’s Robin van Persie.
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