Boro had agreed a £2m fee with Crystal Palace for the midfielder and gave him a medical after showing him around Teesside on Thursday.
Southgate told the Evening Gazette: "It's disappointing to see Ben go to Wigan because we've put a couple of months into trying to make it happen.
"I think he would have slotted in well here and I'm sure he would have enjoyed himself.
"But we have to move on. We are looking at other possibilities. We have ideas but will keep them close to our chest for the time being.
"The reality of these situations is that by making them public you lose players."