The 35-year-old's contract ends in June and he has been linked with moves to several clubs in Europe.
Southgate told BBC Tees: "We have decided that we would make him an offer and we are in negotiations with him but I'm aware he has got other options.
"If we can keep Mark it would be good, his experience is important for us."
Schwarzer joined Middlesbrough in 1997 but Southgate believes he has plenty of talent waiting in the wings should the Aussie choose to leave.
"He might feel that he has been a fantastic servant for the club and is ready for a change himself," Southgate added.
"In the end Mark has to want to stay here. If he doesn't, I've got two very good young goalkeepers and it's maybe the time to give them their opportunity."