The rumour stepped up a notch at Hockenheim, where he drove for the Swiss team in Friday practice.
The Italian is a member of Ferrari's driver academy. This year, Ferrari's Charles Leclerc is at Sauber, but he is linked with a move to Ferrari for 2019.
"We will discuss the issue with Ferrari and then make a decision," said Sauber boss Frederic Vasseur at Hockenheim.
"It's important for Alfa Romeo that we achieve good results with two drivers who are best suited for our future."
Giovinazzi, 24, said at Hockenheim: "It was nice to finally be back in the car, but everything felt normal.
"I know the team well now, since I did my first two races with them in 2017. Since then, Sauber has grown as a team so it would be silly to say I don't want to race in formula one with them in 2019."
When asked about Giovinazzi's comments, Vasseur said: "It's nice that Antonio wants to drive for us. It's something to be proud of.
"He will work with us in several practice sessions and I can get to know him better and find a common language. Last year, we did the same with Charles."
Sauber are ninth in the constructor standings ahead of today's German Grand Prix.