The Nottingham Forest midfielder is viewed as a prime summer target for both Manchester giants.
Forest rate Anderson in the £100m class, while City and United hope to do business for closer to £75m.
The Mirror says both Manchester clubs will push to prise Anderson away from Forest before this summer's World Cup.
For their part, Forest may wish to wait until after the tournament in the hope of Anderson having an outstanding run with England would increase his valuation.
Meanwhile, the midfielder has assured Forest fans his focus is fully on the club's cause for the remainder of the season.
He said: "It’s been a big two years. I think I’ve done pretty well, to be honest, in the time I’ve been here, on and off the pitch.
"I think I've matured as a person as well. Thanks to the club, really, for giving me that platform, going out there and playing and letting me do my stuff."