While Babbel, in his first full season as head coach after taking over half-way through last season, said the club still backs him but he must start notching victories.
"We need the results," said Stuttgart sports director Horst Heldt. "I cannot look into the future. I do not know where we will be next Tuesday, in three or four weeks from now."
On Tuesday Stuttgart slumped to a 3-1 home defeat against Sevilla in the Champions League. They have now not won a Bundesliga or Champions League home match since mid August.
Only a few months ago the former Bayern Munich and Liverpool defender was hailed as Stuttgart's saviour, taking over late last year from Armin Veh to steer the club to a surprise third place finish and a Champions League spot.
They were even one of the pre-season favourites for the domestic title.
But their poor form in the Bundesliga, in which they languish in 13th place, and failure to win any of their three Group G matches in Europe, have piled the pressure on Babbel, who is studying to obtain his coaching licence.
"We are making far too many silly mistakes," Babbel said after their defeat by Sevilla. "The overall performance though gave me confidence. I still feel the club's support," he said.
Whether that support will be there if Stuttgart lose on Saturday remains to be seen.
"There is no other way against Hanover. We have to win," said striker Cacau.
Sixth-placed Bayern Munich, on 15 points, host Eintracht Frankfurt, who will be without injured captain Ioannis Amanatidis.
Champions VfL Wolfsburg, in fifth place on 16 points, travel to basement club Hertha Berlin.
League leaders Bayer Leverkusen entertain Borussia Dortmund on Friday while Schalke 04, in third, take on second-placed Hamburg SV on Sunday.
