The Reds sacked Kenny Dalglish following an eighth-place finish in the previous Premier League campaign, despite winning the League Cup and reaching the final of the FA Cup.
But Henry is adamant that there are no demands from him or the board for Rodgers to reach the top four this season despite a three-year absence from Europe's top table.
He told the Liverpool Echo: “We don't want to put any pressure on Brendan by saying 'you have got to make the Champions League this year'. Of course he wants to make the Champions League, we all do and so does every team in the Premier League.
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“I thought it was realistic last season but we were so far off that pace. At this point we're just trying to get things stabilised - that should be one of our goals.”
Henry is delighted from what he has seen from Rodgers so far and the American feels that the club are in a good place to push on under the tenure of the former Swansea boss.
He continued: “Brendan has a very structured approach and he plays the kind of football that we think Liverpool fans will love to see.
“You can see why he was appointed. He's an inspirational figure. He's very charismatic, he's smart, he knows what he wants to accomplish and he's determined to accomplish it.
“Everything I have heard so far in the way of feedback about him has been very positive.”
The 62-year-old took control of Liverpool in October 2010, his first foray into football, and has admitted that he was unsure what to expect at the beginning - but now feels confident in his handling of the club and what the future might hold.
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“We know more now than we did when we bought the club,” Henry explained.
“We didn't know a lot at the start. We have had something like 20 months to study and learn and this is really the direction we want to go in.
“We are working on a long term programme and we are fully committed to this programme. It will be the kind of football we want to play.
“This year what we're really concentrating on is to make sure we're ready for the first league game. That's our number one goal. We have a lot of work to do to instil a new system.”