I have seen this done a few times without consequences BUT always after a pretty strong informal expression of client's interest.
You will need support from client company anyway, so I would start with an informal conversation with your team there. If that team wants you they can cut most of the interview process, too.
But there may be pitfalls: they may be contractually restricted from hiring you (not legally prohibited, but enough to make it a pain not worth it), etc. An informal conversation can explore options without your intent to jump ship leaking to your current employer.
You will need support from client company anyway, so I would start with an informal conversation with your team there. If that team wants you they can cut most of the interview process, too.
But there may be pitfalls: they may be contractually restricted from hiring you (not legally prohibited, but enough to make it a pain not worth it), etc. An informal conversation can explore options without your intent to jump ship leaking to your current employer.