The Spain international comes off contract at the end of this season.
The Daily Star says Reds boss Brendan Rodgers wants Alonso, who Liverpool sold for £30m in 2009, back at Anfield.
Sources claim Real would be willing to sell Alonso for a token fee in January to avoid him leaving on a free.
Liverpool have a policy of signing young talent but Rodgers will look for special dispensation from Anfield chiefs to lure back the 31-year-old Spaniard.
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