The Easter Road side are just four points behind third-placed Aberdeen and 10 adrift of Rangers with a game in hand.
Lennon has seen his team beat both sides last month and sees no reason why his newly-promoted team can’t overtake at least one of them in the run-in to the campaign.
He said: “Our aim is now to try and break into the top three with 10 games to go.
“We’re further ahead than where I expected to be at this stage of the season.
“Earlier on, I was thinking we should be capable of getting the top six, but it was a case of ‘how far we can go into the top six, who knows?’.
“Now we’re looking to challenge for second or third place, which is a huge step up.
“At this phase of the season, we are still in with a shout of second place and we certainly have a great chance of finishing third.
“That will be hammered home to the players between now and the end of the season.”
The former Celtic gaffer added: “European football is a realistic target for us and that’s what we must aim for.
,”We can’t win a trophy this season, but I think getting into Europe would be just as good in terms of showing us to be one of the most consistent teams in the country. ”
* Lennon was talking to Hibs TV.