The North East side had been hoping to end up inside the top ten at the start of the season, however their win over Southampton on Monday lifted them into ninth.
Boro have taken 11 points from their last five games and Juninho is hoping the side can maintain the current form and land a top six place.
"We are not aiming for ninth anymore, we are aiming for sixth," said Juninho.
"We are playing our best football for a long time and that is now a real possibility.
"We have five games left and 15 points still to play for, and we want to get as near as we can to 15."
Meanwhile, Juninho has praised striker Massimo Maccarone for demonstrating the kind of form that made Boro part with over £8 million for his services.
The former Empoli man has netted five times in the league this term, but it was his overall display against The Saints that had Juninho purring.
"I am sure the manager is delighted with the way Massimo is playing," he added.
"He has always been a good player, everyone at the club knows that, but he has had injuries which have made it difficult for him.
"If you are in and out of a side because you are injured it is not easy. He had a great game against Southampton.
"He looks very fit and is a very dangerous striker. He has so much ability, that has never been in any doubt."
