The 33-year-old striker joined on a free transfer earlier this week after Wolves successfully appealed when his initial application for a work permit was rejected.
Costa, who was at Chelsea between 2014 and 2017, has not played since leaving Brazilian club Atletico Mineiro in January and Lage says he will need time to be ready.
"Let's see, we have one more day, let's see if he can be available or not," Lage said.
"When Sasa (Kalajdzic) got injured Diego was available, it was a time to talk to him and convince him to come to us, so we are very happy with him.
"We also know he needs time to be ready. He is fit, there are no problems but he needs to be ready to compete at this level.
"Our hope is to get him ready to compete at this level and I think we can bring the work with Costa because that's what we need.
"We need a reference in the box, we need a guy to compete with Raul (Jimenez) and Diego has that profile."
The Wolves boss admitted he tried to sign Costa when he joined the club in the summer of 2021, but the Brazilian wanted to return home.
"I can say now because he is here, when I received the invitation from Wolves, one of the first guys I called was (Atletico Madrid player) Joao Felix to know about Diego Costa.
"Joao said good things about his personality and about the way he continued to play. In that moment he wanted to go back to his country to play.
"If 15 months ago when I think about him in that moment we did not have any chance because he had a different idea for his career."
Costa's arrival eases a striking crisis as Kalajdzic suffered a serious knee injury on his debut against Southampton while Jimenez is still struggling with his fitness.
Jimenez will not be available against City.
Lage added: "Raul is still not available and is still not working with the team. After the time he was in the recovery from the injury, he needed to play three games in a row.
"We talked and he was a little bit tired, sort of like chronic fatigue. Now he needs time to recover. He has had some treatment but he is not training with the team, so he is not available tomorrow."