The Perth club have finished in the top half of the league in each of the past five campaigns.
Saints open their Premiership campaign at home to Aberdeen on Sunday.
"If we were to finish seventh and miss out on the top six I wouldn't see that as a failure for our club," Wright told BBC Scotland.
"Probably our fans expectations get higher and higher each season but we don't really put any demands on the players.
"We want to finish top six if we can and we think we know that over the course of the season we can do that, but we certainly don't say at this stage we will finish top six."
Wright feels the promotion of Rangers, who St Johnstone defeated 3-1 in last season's League Cup, will stiffen competition at the top end of the table.
The Saints boss thinks Rangers will add quality to the league and that Mark Warburton's side could put in a title challenge to champions Celtic.
"If you look at the players they have signed, they have improved the squad in depth," said the Northern Irishman.
"They have a stronger squad than what they had last season. I think with the squad they have assembled they could mount a challenge."
Saints beat Aberdeen twice last season with a 5-1 win at Pittodrie and a 3-0 victory the last time the two faced each other in April.
Wright says this weekend's match, along with their next few games, means they will have to be at their best from the first whistle.
"We have a difficult start to the season in terms of who we are playing," he said. "We have the top three teams in the opening few weeks.
"First up Aberdeen then Hearts in the Cup, Celtic in two weeks with Motherwell a top six side next Saturday. It's a great start for us playing some of the top teams."

The teenager, who is thought to have been trafficked to Russia before ending up in Scotland, was found by a member of the public on 16 January.
He is now in the care of West Dunbartonshire Council's social work department.
Police have appealed to anyone who has information to contact them.
The case emerged at a meeting of the council's community planning management board, according to a report in the Dumbarton Reporter. Police Scotland has since confirmed the details.
Ch Supt Grant Manders told the meeting the teenager had been found on a footpath near Overtoun House.
He said: "A young Vietnamese boy - 16 years old - came into our care having been trafficked from Vietnam to Russia.
"Smashing wee boy escaped his traffickers up in West Dunbartonshire and was found cowering in the bushes having escaped from wherever he might have been going."
It is thought the boy ran for about an hour from the site of his escape before being found.
Det Con Andy Docherty said: "He was found in quite a distressed state.
"We quickly established that he was of Vietnamese origin and he stated that he was subject to a human trafficking. His welfare is paramount.
"Inquiries are ongoing."
Manager Tommy Wright says St Johnstone once again have ambitions of a top six spot but realises competition will be fierce this season.
[NEXT_CONCEPT]
A 16-year-old Vietnamese boy was found "cowering" in bushes in Dumbarton after escaping from human traffickers, police have said.