Morbidelli and Quartararo battled with tricky and mixed wet conditions at the Ricardo Tormo Circuit to finish ninth and 11th respectively on Saturday.
Morbidelli managed his best time of 1:41.557 as the dry line started to form while Quartararo continued to struggle and only came in 11th fastest in 1:41.943.
Championship leader Joan Mir of Spain has the slight advantage of returning the fifth fastest time of 1:40.704 on his Suzuki bike with compatriot Pol Espargaro starting from pole for KTM with 1:40.434.
Frenchman Quartararo needs to play catch-up as he is trailing 14 points behind behind Mir, who leads with 137 points.
Morbidelli is also in contention for the championship title if the Italian can overcome a 25-point deficit with three races to go.
“We chose the soft tyre option and worked to get the maximum out of them, so ninth today is not too bad.
“If we make a good getaway on the line, we can move ourselves higher in the opening lap.
“It was a strange day with the conditions changing constantly but it was good. I think we can do a good job in the warm-up and then again in the race,” said Morbidelli.
Quartararo is hoping for better fortunes in the race.
“We have tried many things with the bike in the wet conditions this weekend, to try to understand where we can improve and find some more speed.
“At the beginning of the session, we had a similar pace to those around us but as the laps continued we did not make the improvements that we wanted to.
“We need to analyse the data to try to understand why but I am hoping for better weather during the race,” said Quartararo.