The Saints’ home win over Hamilton on Saturday took them level on points with the Jambos who on Sunday were hammered 4-0 by leaders Celtic.
Hearts have won only one of their last seven league matches but head coach Ian Cathro has revamped his squad, with Greece international Alexandros Tziolis confirmed as the Gorgie club’s seventh January signing on Monday.
Wright does not expect anyone to come or go out of his club before the transfer window closes on Tuesday night, and ahead the game against Partick Thistle on Wednesday night he said: “We are not going to get over confident.
“For to be competing with a club that size for fourth place, if you compare the two clubs, the money they have spent in the summer, the money they are spending now ...so in terms of what they are able to recruit and bring in, you would have to say they were favourites for fourth place but we will wait and see what happens.
“I don’t look at it as us and them, we just try to do the best we can and see where it takes you.”
While looking to overtake Hearts, Wright is aware that victory at Firhill can take his side 11 points ahead of their sixth-placed hosts.
He said: “They have good players and they are always difficult to play against.
“They keep the ball particularly well and sometimes you have to be patient and disciplined.
“But we go there knowing a win can put us 11 points clear of them which is a massive incentive for us at this stage of the season. We can put ourselves in a really strong position.
“We are coming into the stage of the season where we have been strong in the last few years and hopefully that will be the case this season because being level with Hearts, that fourth place could get end up getting you a European place this year with the way Celtic are going, and of course we are still in the Scottish Cup so there is a lot to play for.”