Henry, Liverpool's principal owner, said before the start of the Barclays Premier League campaign that although a title challenge was unrealistic it would be a 'major disappointment' if the club does not reach the top four by next May.
Despite being encouraged by the ?48.5million summer revamp of the squad, Reina, 28, also feels it is too much to expect Liverpool to challenge champions Manchester United.
Hot pursuit: Reina says Liverpool must break into the top four
But the Spanish goalkeeper believes it is vital Kenny Dalglish's squad capitalise on not playing in Europe this year by improving on last term's sixth place.
'It's difficult because we have been some distance from those aspirations for a long time,' said Reina, who reaffirmed his commitment to Liverpool last season after being the subject of interest from United and Arsenal.
Stark warning: Henry
'The priority is to return to the top four and keep improving. It's fundamental the team gets back into Europe, then next year makes the jump to challenge for the title.
'I have always been in the situation of playing in midweek so it's new for me. They say it can be an advantage (not playing in Europe) and I hope we can benefit from not having such a heavy schedule.
'I think the new signings have the advantage of having played many years in this league, that always makes things easier.
'This club has a big history and we have to follow those legends.'
Reina also defended David de Gea after his compatriot's error-strewn start at Old Trafford and insisted the ?17.8m goalkeeper will live up to his price tag.
'It is typical that the critics smell blood when there is a mistake,' said Reina. 'But if Mr Ferguson has put his trust in him by signing him then it seems logical that he will settle and be a quality signing for them.'
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