Aéropostale Inc. may have to put its chain of teen apparel stores up for auction by mid-July, but it is holding on to hope of a turnaround and battling its top lender, an affiliate of private-equity firm Sycamore Partners, for a chance.

A brewing bankruptcy-court fight between Aéropostale and Sycamore Partners—as outlined in court documents—focuses on the crucial back-to-school selling season, normally a high point for the retailer’s revenue.

Sycamore wants Aéropostale to pick a lead bankruptcy auction bidder by July 1 so buyers, possibly including liquidators, get the advantage of taking over when students hit the stores for their fall wardrobes.

Aéropostale says the money that will come in from back-to-school sales could help with a reorganization or attract a buyer that will keep the chain in operation. The alternative is a liquidation, which threatens to wipe out thousands of jobs and could leave hundreds of stores dark.

The retailer filed for chapter 11 bankruptcy protection May 4, closing some stores and hoping to save the rest. That hope, Sycamore said in its filings, is “illusory,” and “has no realistic chance of success.”