The French giants currently have Jose Anigo as caretaker boss following the winter departure of Elie Baup.
And Labrune admits they were in discussions over a deal with the former Chelsea, Tottenham and Porto boss.
He told L'Equipe: "Andre is a very, very high level coach.
"Our talks and his words about future of l'OM confirmed it to me. One month ago, the ball was in my court. He accepted our efforts but it was based on a financial arbitration and was bad timing."
Villas-Boas has since agreed to take charge of Zenit, and Labrune admits that Marseille are now turning their attention cutting their squad size in order to get their finances back in order.
"We have to make some money selling players before June 30 to take get our accounts on the right track again," he admitted.
Labrune, though, insisted they would not be selling some of their best young assets including Florian Thauvin, Benjamin Mendy, Mario Lemina or Brice Samba.
"For each of them I received offers superior to the fee we paid for them, but we rely on them for next season," he said.
"They won't be the same anymore as they will master a bit more of the l'OM environment. Thauvin? His value on the market is very high but it's out of the question to sell him."
He did admit he was less sure over the long-term futures of Andre-Pierre Gignac and Souleymane Diawara.
"The current problem is not about who will leave and who will stay, but to have results that match our ambition, because the composition of our group will also depend on our sporting situation," he said.
Marseille are currently in sixth place in Ligue 1, some nine points off third place and the final UEFA Champions League spot.