The Chilean international is on the brink of leaving Arsenal for Old Trafford, with the transfer expected to go through at some stage on Monday.
However, Ince has suggested that the likes of Romelu Lukaku and Anthony Martial will now fear for their place at United, despite the pair combining to net United's only goal in Saturday's 1-0 win over Burnley.
The 50-year-old told The Mirror: "I would imagine there are some players who are panicking about his arrival. Mourinho is clearly looking to improve and change that team up, to make them more enjoyable to watch. That means people will need to make way if they can't meet the standard.
"You know what you're getting from Romelu Lukaku. Anthony Martial can do that, when he switches on, which isn't often. Marcus Rashford is still young and doesn't get the game time, but none of those players are on the same level as Sanchez.
"None of them have that wow factor. They're good players on their day – but Sanchez is a game changer. He's priceless."
Sanchez could potentially make his United debut in Friday's FA Cup tie with Yeovil Town.