The Everton defender is set to join on a season-long loan deal.
The Magpies have been busy this summer, adding no fewer than eight players to their squad so far.
Manager Rafa Benitez’s business is not finished there though and he is still looking bring in more new faces before the end of the transfer window.
He wants to add competition for Paul Dummett at left-back and that has seen him targeting Everton defender Galloway.
The youngster impressed for the Blues last season, but he is facing a fight to get into the team under Ronald Koeman with Leighton Baines and Bryan Oviedo challenging.
As a result he is set to sign on a season-long loan deal with Newcastle according to the Chronicle, who will pay a substantial loan fee as well as his full wages to secure his signature.
They report that Benitez has targeted him because he is quick, athletic and able to support attacks, something the Spaniard is looking to add as he attempts to bring more balance to his side.