Off contract in June, Monreal is keen to stay and play on with La Real.
Mundo Deportivo says the veteran wants a two-year contract to re-sign - and La Real are keen for the partnership to continue. Monreal would be 37 should he play out those two years.
Meanwhile, talks over a new contract with Adnan Januzaj are dragging on.
The Belgium international has a deal to 2022 and his agent Mino Raiola met with La Real president Joakin Aperribay in Turin a fortnight ago.