The Sun says Moyes failed with a £12m bid for his former left-back yesterday.
But Moyes clearly knows what it will take to lure the 28-year-old away from Goodison.
And it is understood Everton will be tempted to do business if United up their offer by another £4m.
The move also casts doubt over Patrice Evra’s future after 7½ years at Old Trafford.
Evra, 32, has been linked with moves back to his native France with moneybags Monaco or Paris Saint-Germain.