"Mark's at the end of his current contract and we decided that we'd make him an offer," Boro boss Gareth Southgate told the club's official website.
"It was difficult to do that when we didn't know what division we'd be in.
"I had to think long and hard about whether I wanted Mark to stay and whether that would impact on the other goalkeepers we have at the club. But if we can keep him it would be good, his experience is important to us.
"We're in negotiations with Mark. But I'm also aware he has other options as well and he might feel that he's been a fantastic servant to this club and he's ready for a change himself.
"From January he's been able to speak to other clubs across Europe. I've a fair idea how far down the line those talks are, but we wanted to give him something to think about in terms of a contract offer from ourselves.
"But in the end Mark has to want to stay here and, if he doesn't, I've got two very good young goalkeepers and maybe it's time to give them their opportunity."