The midfielder is due to cost the promotion-chasing Ashton Gate side around £1 million.
Walsh, 20, is about to end his time at Goodison after returning from a loan spell at Birmingham City.
City boss Lee Johnson has been given the go ahead to bid for Walsh.
And he has already met with his advisers ahead of a swoop early in the January window.
Walsh faces fierce competition to get into the Everton side.