The Mirror says Barry, 28, is about to enter the final year of his £48,000- a-week contract at Villa Park.
The England midfielder has stated he will stay if Villa, who are fifth in the Premier League, can offer him Champions League football.
But Liverpool, who failed to sign Barry with four bids last summer, are still interested.
O'Neill has pledged not to stand in Barry's way should he leave after pleading for another season last summer.
But O'Neill is still pulling out all the stops to persuade Barry he can fulfil his ambitions at Villa Park.
The pair were at a club dinner this week and O'Neill, whose side host Everton tomorrow, made special mention of his captain.
"Young Gareth has a heart the size of a lion," said O'Neill. "He wants to play every single week and wouldn't take a break if you asked him.
"He doesn't get a break in the season and plays international matches. It is great for him, absolutely fantastic and he is very, very talented."