Throughout pre-season there was a constant suggestion from within the Magpies camp that just finishing outside the bottom three would suffice in their first campaign back in the Premier League.
However, after winning four and losing four of their opening games in the topflight, there is a gradual change in focus within the dressing room.
Last Sunday's 5-1 success over their arch-rivals lifted Newcastle up to seventh ahead of tomorrow' s trip to second-placed Arsenal.
And while there might be a belief that they will find it difficult to return to Tyneside with anything other than a defeat, Simpson does not think that a top ten finish is beyond Newcastle.
Firstly, we want to prove that we are back as a Premier League club and anything else is a bonus, he said. At the start of the season, we'd have taken 17th because that would have meant survival.
At the moment we are seventh so I'm not saying we would take 17th right now.
The way we think now and the way we have performed, we know we can do much better than that.
We are not getting carried away with seventh place. You have got to be realistic. We got out of the Championship and we are in a much tougher league.
The way some of the players are performing, we know we can end up a lot higher than 17th. We have talked about the highs and lows of a season. If you lose two games, you are right there at the bottom again as it is so tight. That is the way we are looking at it, we can't look too far ahead.
Arsenal's exceptional style of football, particularly at home, will make it difficult for Newcastle to claim a third successive Premier League win tomorrow.
But Simpson, a former Manchester United full-back after graduating through the ranks, is not ruling out another surprise victory, knowing how West Brom emerged with three points on September 25 from North London.
We go to Arsenal with a positive mentality that we can get something there, he said. We are capable of beating a top four side and then losing to a team in the bottom eight.
That is what we have learnt especially after the Aston Villa game. We then lost at home to Blackpool.