Given is currently on loan at the Riverside from Aston Villa, as he has been covering for Jason Steele, who has just served a three-match suspension.
However, Steele now has an ankle problem so Boro are hoping to extend Given's stay.
"We will be having discussions with Aston Villa," said Boro boss Aitor Karanka.
"It depends on what Shay wants, and on what Aston Villa want to do, but I would like to have him here for the whole of the season. That would be my choice, but it is not just up to me."
On Steele, Karanka added: "Jason is not training at this moment.
"He has an injury problem with his ankle, and the doctors are working with him. I think it is a long-term thing, but the doctors are working hard to try to make sure that Jason can be back with us as soon as possible."
Swansea are now being strongly linked with Given, as Michel Vorm needs surgery, but Michael Laudrup says they have not made a move for the Villa man - although he is a fan.
"I don't think he (Given) is on our list right now but he is a talented keeper," said Laudrup.
"There are a lot of scouts in the club looking at a lot of players but we are really happy with the two goalkeepers we have now.
"It is certain that Michel has a problem, but it's not a big problem. He needs a bit of surgery but we are not talking a long time to recover.
"But you have to be up to date with positions in your squad as you never know what might happen in five or six months time. But it's more about having a list (of targets) as in a few weeks the window starts."