The Sunderland Echo says United are reported to have made a bid of between £500,000 and £1m for the highly-rated 18-year-old.
Newcastle have made no comment on the reports linking the club with Bigirimana, and City manager Andy Thorn - who had a spell at Newcastle as a player - has insisted he wants to keep him at the Ricoh Arena.
"Gael's a good talent, but he's still got a heck of a lot to learn, and I'd be really disappointed if he left," said Thorn.
"I was a young lad when I went to Newcastle. They're a massive club, and if you aren't careful, it eats you up if you go too soon, so you have to be very careful.
"He was given his chance last season. Then, like most kids breaking through, he had a little dip, and we looked after him by taking him out of the team to get him back on his game again.
"He understood and did that, and he came back strong again, so it would be a real shame if we lost him now.
"He's very much part of my plans, and I'd love to see him continue his development with us."