Welsh could be tempted into leaving Sunderland, according to the Sun.
Sunderland's development squad beat Manchester United 2-0 at Old Trafford yesterday.
Welsh feels Moyes has been a positive influence on younger players at the club.
Sunderland's under-21 side beat Middlesbrough 1-0 yesterday evening.
The Frenchman has not featured in the first-team since August 2014.