If I don't include my previous manager's name for a current job, can they verify if I got fired?

I was fired Company A - Fortune 500 and hired by Company B - also a Fortune 500 a couple of months later. The reason of my termination was due to attendance, but had a good work performance. When asked by HR why I'm no longer employed by Company A, I said I was let go due to restructuring/downsizing (which is currently happening, just not in my case). If I don't include my previous manager's name, and provide a prior co-wkr for reference, is there any other way that Company B discover the real reason I'm no longer employed?