The 21-year-old was a member of Boro's 2004 FA Youth Cup-winning side and made his first-team debut, aged 17, the same year.
But he found it difficult to break into Southgate's starting XI last term, making just 15 league starts for the north-east club.
And Southgate has admitted Morrison's desire to play a greater role could see him head to the Coca-Cola Championship club.
"He's down at West Brom having talks and they've agreed a fee with us," he told club website.
"It's a difficult one because with young players you want them to come through into the side - at the moment we are just waiting to see what he wants to do.
"It's whether there's a better guarantee of first-team football for him."
Southgate added: "We will know more over the next couple of days as to what he wants to do."