The Costa Rica international was linked with the Premier League giants last week - and their interest remains, says Diario Gol.
Keylor is weighing up his options at Real, fed up with the constant rumours of big name keepers arriving to replace him.
The Costa Rica international, despite the support of coach Zinedine Zidane, could jump before being pushed - with the Premier League a favoured destination.
Arsenal and Liverpool are both monitoring his situation, with Keylor having asked former Costa Rica star Paolo Wanchope about life in England. The former Ticos coach played for Manchester City and West Ham.
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