Boro turned down a £4.5million bid from Portsmouth for O'Neil last week, and Southgate has praised the midfielder's attitude.
Southgate: "We fully understand Gary's situation and he understands ours.
"We know that Gary loves playing football here but he has become a little bit frustrated because his family are unsettled.
"We have asked him to do a job for the club over the next few months and then we will see what the future holds."