Both of Boro's keepers are out of contract at the end of the season and would be free to sign pre-contact agreements with rival clubs in January.
Turnbull has played in 17 of Boro's 19 games this season.
Southgate: ""We're pretty close with both. We'd hope to have something to announce formally in the next couple of weeks.
"Brad's pretty much done, he realises what I think of him as a keeper.
"Also, if he goes somewhere else he's probably not going to go as first choice because people are going to want to see him play.
"So he would face the same battle as he's got here and he maybe thinks that rather than taking on an experienced No 1, there's still a very level fight on their hands here.
"Ross has been fantastic but I picked Brad on the opening day and that was for a reason, so the challenge will go on between them."