It would be best if you could wait for your supervisor to come back from leave before starting any transfer process. If the other supervisor is keen on you, they should be prepared to wait.
When your supervisor gets back, you should ask their permission as a sign of respect before engaging with the other supervisor with regards to a job switch. In a perfect world you wouldn't need to do this, but you all work in the same company, the last thing you want to happen is to create friction. The conversation should start off around your ambitions to grow as an employee.
They may respond in a few different ways:
They may ask you not to apply
Would be unusual, but you can start a discussion with them about how you can satisfy your career objectives. You can, of course, apply anyway.
They give you their blessing
It's full steam ahead
They warn you against the move
You then can work through the different issues with your supervisor and see if the move is still right
They may offer you advancement opportunities within the team
If they understand your ambition, they may open up and advancement oppertunity within your current team. You can then decide if that is right for you.