The Red Devils sit fifth, two points behind Arsenal after their loss to the Gunners at the weekend.
That was Ole Gunnar Solskjaer’s first Premier League defeat since he replaced Jose Mourinho.
There are just four points separating Tottenham third and Chelsea in sixth with Liverpool and Manchester City clear in the race for the title.
And Young feels Manchester United should be far higher than their current position.
When asked if the race for the top four could go to the last round off Premier League fixtures, Young explained it could well do.
He said: “Yeah, of course, but I think there are so many points to pick up, so many games left.
“For me, if we were talking back in October, saying that we were going to be fighting for fourth place, we would have taken that with where we were.
“But, on a personal level, and everyone in the dressing room, I think the club a size of United should be higher up the league and that’s where I see it.
“But right now, obviously, we’re talking about finishing in the top four and we’re going to do everything we can to do that.”
United do not return to Premier League action until March 30 when they welcome Watford to Old Trafford.
Before that they face Wolves in the quarter-finals of the FA Cup on Saturday (7.55pm).
Young is out of the match as he serves a one-match suspension in the competition.
Arsenal could potentially be favourites to secure a place in the top four as they face none of the top six in their last eight matches.
Chelsea, Tottenham and United will all look to ruin that idea though as Unai Emery looks to end his first season at the Emirates on a high.
The teams that finish in fifth and sixth will have to make do with a place in the Europa League and some tricky trips on Thursday evenings.