Since Ole Gunnar Solskjaer replaced Jose Mourinho the Red Devils are unbeaten in 10 matches.
And that has seen United move up to fifth in the table, two points behind Chelsea, who occupy the final Champions League qualification slot.
On the race for a top-four spot, Phelan said: “We’re chasing a pack but I have known Manchester United claw back points before.
“We can only do our own job, keep doing what we’ve been doing, keep pressing all of the right buttons.
“We’re going to lose a few games, which happens in football, but at this moment in time we’re winning a few as well.”
Despite the improvement at United since Solskjaer replaced Mourinho, Phelan wants to see even more from the players.
“Our aim is to try and do even better, to try and push the players even further,” he said.
“I’ve experienced a lot of things at Manchester United and I want this next generation of players to do the same.
“They’ve won some things, but they haven’t won everything – and I want them to win everything. Why shouldn’t we?
“You’ve got to be greedy in this game and to keep pushing. What they’re here for, surely, is to be better players, better individuals and to win something for the club and themselves.
“Players are talented here because they’re here for a reason. It’s then up to them to embrace Manchester United, the ethos, and everything around it. That’s the reason why they’re here.
“Ability shouldn’t be an issue – they shouldn’t get through the door if they haven’t got the ability, so what is important is that we make sure we only bring in that quality and personality, that charismatic sort that can enhance what is already here. That’s a major responsibility.
“Winning is there, it’s got to be – the size of the football club demands that. We’re trying to build something.”
Phelan also says it is vital United players express themselves and uphold the traditions of the club.
“You try to bring a smile to people’s faces, and try to tell people of the adventure you’ve been on, the success you’ve had and what it takes,” he explained.
“It doesn’t happen overnight, it’s something you have to work at, but you need people to make it happen, people who enjoy this environment and enjoy coming into work.
“I think that’s massively important. If that’s happening with the staff, it transmits to the players.
“Then they get on to that football field and really look forward to playing and showing their abilities to the huge amount of supporters we have around the world.”
United return to action on Saturday lunchtime against Fulham in the Premier League (12.30pm).
A win would see them move into the top four before Chelsea take on Manchester City on Sunday.