It's very possible that the glue used for this was hide glue. Hide glue is water soluble, particularly for hot water. You can try to use an iron on a damp rag to force steam through the wood and this will possibly reactivate the glue. But hide glue is really weird stuff and I have found it to be pretty unpredictable under most circumstances. The amount of existing finish also makes this a problem to predict.
I'm not sure exactly where you want to end up with this, a new layer of veneer or no veneer. Remember that veneer is also used to cover a more expensive wood with a cheaper one. Veneer is available today from any sources and it actually is thinner than it was years ago. Replacing the veneer and careful sanding are possible techniques.
Honestly, if you really like the table, you should take it to an expert. There is a good bit of art in working with veneer.