Bruce famously never won an England cap, despite winning 13 major honours across spells with Norwich City and Uniyed.
“Brucie was awesome," Pallister told the One Last Match podcast.
“It was [Eric] Cantona's first game he played for Manchester United – he came on as a sub. Afterwards, Bobby Robson came to our hotel where we were having a couple of jars in the bar after the game, and he walked in and said to the manager: 'Can I have a chat with your boys?'
“The gaffer introduced Sir Bobby. He said: 'Look lads I just want to have a moment of your time. I just want to say sorry to Steve Bruce because I really should have given him an England cap. You were so well worth an England cap - I just want to apologise for not giving you one.'
“Which was great for Steve to hear that. It didn't probably ease the pain of not playing for England, but I think that told you an awful lot about what really great managers thought about Steve Bruce."