Boss Gareth Southgate, however, is still hopeful of persuading goalkeeper Schwarzer to sign a new contract at The Riverside.
His contract runs out this summer and he has so far refused to sign an extension, prompting reports of interest from a host of European clubs.
Southgate told Boro's website: "We are talking to Mark. I'd like him to stay. His experience is important but he has real drive and determination that is still increasing.
"He is the one in our dressing room that can truly say he has played at the highest level - in World Cups, in Europe, in cup finals and so on.
"We have to negotiate and see where we go, but players have to want to stay here and we have to want to keep them.
"Mark has been here a long time so he has to decide if he needs a fresh challenge, whether what we are trying to do is going to excite him, whether he feels he is going to be appreciated here."
