Ferguson apparently sustained the injury in a first-half clash with Samir Nasri during the 2-1 victory over Arsenal, but remained on the pitch throughout the 90 minutes.
"Barry Ferguson played an hour with a broken rib at Wembley," McLeish said. "It shows that his pain threshold is frightening and also the character of the guy to do that. We all know how painful a broken rib can be.
"I think it happened in a challenge with Nasri. He has taken a right bash in the ribs. It has been confirmed that there is a stress fracture and he is a doubt for the weekend."
Should he fail to prove his fitness for Saturday's game against West Brom, McLeish suggested he could take a pain-killing injection to avoid missing further games.
"Ferguson has had this before in our time at Rangers and he took an injection into the ribs," he said. "It was administered with a big giant needle - not something I would fancy - which alleviates the pain.
"If he doesn't make it for the weekend, that could be administered in the next week or two."
McLeish - who also revealed Craig Gardner is doubtful for the weekend - said he wanted to use the Carling Cup victory to help attract more quality signings over the summer.
"I think this achievement can be a spur," he said. "If we continue to attract the right players, with the right character and determination, then you are always going to have a chance.
"The quality factor is very important to me. We really need to concentrate on bringing the quality in, but you just can't bring in a player of quality if he doesn't have the determination and desire that this group of players have.
"They have a fantastic spirit and never-say-die attitude."