The paper models fixed rate tenders, where a central bank offers to lend central bank funds to financial institutions. Bidders are constrained by the amount of collateral they have. We focus on the strategic interaction between bidding in the tender and trading in the interbank market after the tender, where short squeezes could occur. We examine how the design of the tender affects equilibrium bidding behavior and the incidence of short squeezes. Important elements in the analysis include the type of policy implemented by the central bank as well as bidders' initial endowments of liquidity and collateral. Three instruments for softening short squeezes are identified: the tender rate, the tender sizes, and admissible collateral. Increasing the tender rate or size tends to decrease the probability and severity of a short squeeze. The possibility of a short squeeze may induce bidders to oversubscribe even if the tender rate is higher than the competitive rate.