The 22-year-old's stock has risen following his curling strike from outside the box in France's 4-3 knockout win over Argentina at the World Cup on Saturday.
However, Reschke told Stuttgarter Nachrichten that the Bundesliga outfit want Pavard for at least another season.
"Benjamin Pavard was already in the focus of the big clubs as a Bundesliga player," the director said. "That has now increased once again. We want to keep the player - let's wait, what the next weeks and months bring.
"We will definitely not sell Benjamin Pavard this year for 50 million euros, but we will do everything to ensure that he plays with us for another year.
"We are ready to find a solution for July 1, 2019, with a club that works cleanly and correctly with us."
Pavard, who joined Stuttgart from Lille in 2016, is contracted at his current employers until 2021.