Valverde's current deal with the Catalan outfit runs through until the end of the season, but according to Spanish publication 'Sport', he is in no rush to extend his stay at the club and will instead wait until the end of the season and assess how his side does before committing to staying on or not.
The 54-year-old took over the club in July 2017 and won a LaLiga and Copa del Rey double during his first season at the helm, going agonisingly close to completing a whole league season without defeat.
The reports suggests that Valverde feels less secure in his role following the departure of Robert Fernandez as the club's technical director, but that the former Athletic Bilbao boss maintains the full respect of the dressing room.
It is understood that Valverde's current contract contains an option for a extra year, though it will have to wait for the end of the season to be activated if it is at all.
The LaLiga champions are without a league win in four games, having suffered a mini-wobble, with Sevilla leading the way in the Spanish top flight.