The defender produced a superb goalscoring performance in France's World Cup win over Argentina on Saturday.
Before the World Cup kicked off, the youngster said: "I cannot promise (staying). Because if I promise to stay in Stuttgart and not do it then the fans in this world will not like me anymore. But I want to play Champions League football, sure. I do not hide that.
"My advisor and I will take our time. Now it's time for the World Cup. We have our ideas.
"We do not want to be pressured and decide quickly."
Meanwhile, Stuttgart chief Michael Reschke revealed to BILD this morning: "The topic of Pavard has picked up at even more pace since Saturday. There is interest and requests from top clubs in Europe. But we would not sell Benji for €50 million this summer. We want to keep him. "
Pavard has a buyout clause in 2019 for around €35m.
Reschke added: "We are interested in finding an early solution with one of the interested clubs for the 1st of July 2019. The decisive factor is that Benji also wants that."