Marcus Rashford was the latest player subjected to racial abuse, following similar treatment of team-mates Axel Tuanzebe and Anthony Martial.
“A number of these comments have been reported to us and we are liaising with those involved to provide support and we will be investigating these crimes thoroughly,” said a GMP statement.
Rashford described the abuse as “humanity and social media at its worst”.
The 22-year-old, who was awarded the MBE for his work fighting child food poverty, received multiple racist messages on Instagram on Saturday.
They were sent to the England striker after United’s 0-0 draw with Arsenal.
“I’m a black man and I live every day proud that I am,” he said on Twitter.
“No-one, or no one comment, is going to make me feel any different. So sorry if you were looking for a strong reaction, you’re just simply not going to get it here.
“I’m not sharing screenshots. It would be irresponsible to do so and as you can imagine there’s nothing original in them.
“I have beautiful children of all colours following me and they don’t need to read it. Beautiful colours that should only be celebrated.”
It added: “Nobody should be subject to such abuse and it is deeply upsetting not only to those who suffer it, but to all those who come across this awful language too.
- Nobody should be subject to such abuse and it is deeply upsetting not only to those who suffer it, but to all those who come across this awful language too. These hateful words have no place anywhere in our society whether online or otherwise
“These hateful words have no place anywhere in our society whether online or otherwise.”
Facebook, which owns Instagram, also released a statement after Rashford’s abuse. It said: “We have taken action in this case by removing accounts and comments and are continuing to investigate.”