The nine-time European Cup winners have sounded out the Liverpool captain's agent Struan Marshall several times and believe the player is open to the move, says daily sports paper AS.
However, Madrid are reluctant to pay anywhere near the mooted £25million asking price for a player who recently turned 30.
Mourinho wants to add experience, but Bernabeu president Florentino Perez has made it clear they cannot afford to match last summer's lay-out of more than £200m.
Interest in Ashley Cole has cooled because of the high fee that Chelsea would demand, with a move for Lazio's Alexander Kolarov - a Manchester City target - thought to be imminent.
Madrid are also in a strong position to sign right back Maicon from Inter Milan and winger Angel di Maria from Benfica, leaving the central midfield position as their most pressing headache.
They have put aside plans to sign Daniele de Rossi for now because of Roma's insistence that the midfielder is not for sale and they will not pay over the odds for Gerrard.
Valencia's David Silva has been presented as an alternative - as has the fanciful option of signing Cesc Fabregas from Arsenal - while Athletic Bilbao's Javi Martinez, Inter Milan's Thiago Motta and Bayern Munich's Bastian Schweinsteiger are also in their sights.