And he also wants to tie down Ashley Young, Gabby Agbonlahor and John Carew on lengthy new deals.
It will mean smashing the club's current wage structure.
Barry, on £50,000 a week, is the highest earner at Villa Park but could double his basic pay at Anfield.
But American chairman Lerner met O'Neill this week to discuss strategy. And the Ulsterman revealed: "Randy is well versed in these things and it did not take a long conversation to find he shares my ambition to keep Gareth.
"We want to build a side here but it should not surprise me now that players are being mentioned with the big four.
"If it's not Gareth this week it will be Ashley next week and Gabby the week after. Maybe John Carew.
"I will give everything in the summer every consideration. We would naturally address the size of the squad. But I want to start from the position of the players who have done fantastically well here and Gareth Barry is one of them. There are others who I want to keep at the club who I believe should have extended contracts.
"But it is a fact that top-quality players want top-quality football and that means the Champions League. That's what we have to aim to get to."