The Salentini, who host Palermo, are languishing in the relegation zone, where they have spent the whole year.
However, Cosmi will not accept his side have already flunked the season and believes there is still a lot to play for.
"I still think a lot can happen," he said. "Eleven games are a lot and the signs I am receiving from the team give me reason to hope.
"We can defend our Serie A status but we have a lot of work to do and we need to start climbing up in the standings."
Lecce saw their five-match unbeaten run come to an end in last weekend's 2-0 loss to Serie A leaders AC Milan.
The result left Lecce in 18th place, six points from safety.
Lecce will be bolstered by the return of defender Nenad Tomovic, midfielder Gennaro Delvecchio and striker David Di Michele from suspension.
Palermo travel to the Via Del Mare stadium looking for their first win away from home this season.
The Sicilians have mustered just four points - all from draws - in their 13 games away from home.
Last weekend's 1-0 defeat to Roma was Palermo's third straight loss and dropped Bortolo Mutti's side to 10th place.
Lack of goalscoring has been Palermo's main problem of late.
The Sicilians have not scored in their last two league games.
"We must do better," Mutti said. "We must create goalscoring opportunities and make the most of our chances."
To complicate matters for Mutti, he will be without the team's top scorer Fabrizio Miccoli, who serves a one-match ban.
Defender Eros Pisano and midfielder Edgar Barretto are also suspended.