The winger, 20, was stalling over the terms of a new deal but Barca are now on the brink of securing the player boss Pep Guardiola regards as the long-term successor to David Villa on a long-term agreement.
Spain Under-21 international Cuenca said: "My deal will end in June, but my plan is to remain at Barca.
"My desire is to sign a new contract, and talks are very advanced. We are almost there.
"For me to be in this Barcelona team is a dream."