Dumbledore had truly cornered Oz with this proposition. At first glance, Oz would be inclined to decline Dumbledore's offer of tutoring and go on his merry way however Dumbledore was known as the second coming of Merlin for a reason. Learning under such a man privately could allow Oz's magic prowess to improve tremendously in the years to come.
Also since the man is directly connected to Merlin in some ways, there's a chance he will know more about Merlin than Oz does at this point. Such information is invaluable to Oz. There's even a chance that the man has procured some of Merlin's artefacts over the years.
In Oz's mind this proposition was like a piece of meat at the end of a stick. It was too tasty to pass up, yet he knew he was being led along.
'If I accept now Dumbledore will immediately send out reports and news that I'm officially learning under him. That would in turn cushion the amount of damage due to the Harry incident and also improve his stature even more as someone who is leading the wizarding generation to a new era via me, a descendant of Merlin... This old man indeed got me good here...'.
Looking at Dumbledore who was giving an innocent grandfatherly smile, Oz sighed.
"Okay Dumbledore, Your proposition is indeed enticing. I have a question though before I decide whether or not I accept".
Dumbledore's eyes narrowed as he looked at Oz.
"And what would that question be Oz?".
Oz looked around Dumbledore's office, looking at all the different artefacts strewn about.
"Do you have information on Merlin at all that you'll be able to give to me?".
Hearing this query, Dumbledore smiled.
"I indeed do Oz... as a matter of fact I have a certain artefact that will prove incredibly useful to you, if you so desire to be my pupil".
Oz's eyes flashed at Dumbledore's comments.
"Oh? May I asked what the artefact is?".
Dumbledore looked over to one side of his room and with a flick of his wrist a book came flying to him.
"This is the artefact I have from Merlin... It was actually handed to me by the ministry many years ago to research. Sadly though for me, even though im heralded as the second coming of Merlin I'm not actually connected to the man in any way. This book is guarded by a blood ward that will only accept those of Merlin's lineage. If you come under me, this book will be yours to take".
Oz looked deeply at the book in Dumbledore's hand.
'This book is the book I read about earlier in the year! Merlin's book that he carried around everywhere with him... his own personal tome. If this book does what I suspected and gives me clues on where his abode his, or anything else for that matter it would be an incredibly important item for me going ahead...'.
Oz looked up at Dumbledore's face again, seeing that the old man had a knowing face on.
'First Salazar and now this old man... I keep getting locked into contracts it feels with offers that are far too enticing to pass up. If I denied Dumbledore now, he would probably withhold that book from me for as long as possible, maybe even locking it away somewhere... He knows that what I need now is knowledge of my ancestor and he's using that as leverage. This man truly isn't to be underestimated'.
"One last question Dumbledore, what exactly are you planning on teaching me if I learn directly under you? As you saw in that hallway I've already managed to teach myself the Patronus charm, If self-study can get me to this level in 2 years then what do you have to offer me truly?".
Dumbledore looked at Oz deeply and then leant back in his chair.
"There's many things in this world I can teach you Oz... As you should rightly know, Spells aren't the be all end all, there are many different magical practices out there. One example would be something that I taught to Minerva all those years ago...".
Stroking his beard he started to smile.
"Yes... I could allow you to become an Animagus and train you in that form... seeing your Patronus is a dragon, such a form would benefit you immensely if you pull it off. I could also teach you more about Occlumency and legilimency, although that ring seems to guard your mind well enough for me not to interfere what if that ring is destroyed or something? You need to have your own personal guard be powerful and not rely too much on artefacts. Finally I could also tutor you in duelling... It seems as though with the way you move you've taught yourself with two wands. Indeed thats an admirable feat for sure however..."
Using his open hand again Dumbledore conjured a flame that was hot enough that Oz felt the heat from where he was standing.
"Wands in some ways are a crutch Oz. Having two wands doesn't necessarily mean double the power or double the control. Learning spells wandlessly is where you start to become a true wizard in my eyes. I could teach you how to conjure better wandlessly and implement that in battle".
Oz stood listening to all the things Dumbledore was offering to teach him. As Dumbledore went into more and more detail Oz found it harder and harder to pass this opportunity up. Not only would he receive a tome that belonged to Merlin, but he would also receive teaching in all these subjects he had neglected over the past year or so while training solely spells.
"... Okay Dumbledore you got me hooked. Ill accept your tutoring sessions... on one condition".
Dumbledore narrowed his eyes as Oz was asking for a condition.
"Oh? and what might that condition be Oz"
Oz looked at Fawkes with a slightly evil grin on his face.
"Could you please allow me to have a phoenix tear and feather from your phoenix Fawkes over there?".
Let me reiterate. While it seems Dumbledore is being kinda evil here he isnt an evil person. Hes being extremely manipulative is all which is what we all know him to be.
He truly does care about the greater good hence he sees Oz as an important character, important enough to force to his side via this tactic instead of hoping to turn Oz to his side over time. Or maybe since Oz will be tutored directly under him he will hope to turn him to his side throughout lessons.
As for Oz's final condition, its a test in a way to see how much Dumbledore wants him on his side. If he gives up a feather and a tear to Oz then that shows to Oz his importance in Dumbles eyes. If he doesnt then that shows to Oz theres a barrier between them still.