The ligament that can be compressed between the thumb and index finger when placed in the omental foramen to control bleeding during what is commonly referred to as the "Pringle maneuver" is the **hepatoduodenal ligament**. This maneuver involves applying pressure to the hepatoduodenal ligament, which contains the hepatic artery, portal vein, and common bile duct. It is sometimes used during surgery to temporarily control bleeding from the liver or nearby structures.