No. Sub-tasks belong to issues, and it's the issues that go into sprints.
It's really not a natural behaviour in "business projects". If you're implementing some form of Agile planning, then the sub-tasks need to remain with their parents, and that's how you plan. If you find this doesn't work, your "business project" is paying lip-service to being Agile and really doesn't get how the planning works at all - they should move to Kanban, or skip Agile completely.
Kanban doesn't allow planning because it's aimed at processes that don't need planning. Scrum does planning on a "commit to deliver X amount of stuff in timeframe Y" I suspect your business projects are quite simply not Agile, and you're trying to apply an inappropriate methodology to them. You'll need to analyse what you *really* want from planning (and maybe just add in the use of Kanban boards for nothing more than a bit of visualisation and easy updates)
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