He was also surprised by the offer, he didn't think Tony would want to help someone that almost killed him. But if he was willing... Steve wasn't about to turn him down. Bucky needed his arm to be fixed and at least Stark's words were able to stop him from leaving as well.

"Thank you, Tony." Have a completely sincere looking expression directed at you Tony. He was so grateful and he really owed him for offering to do this still. Of course he planned on staying here during the operation, he wasn't about to leave those two alone with each other. That wouldn't be a good idea.