PREPARE FOR A HUGE ESSAY WRITTEN BY ME AGAIN. IF YOU DON'T LIKE READING STUPID LONG POSTS, THEN GO AHEAD AND SKIP TO THE NEXT ONE.
I don't think Neji would actually show that he wants to learn new moves, like Naruto did. And even if he did, I doubt that Jiraiya would plan to teach him Rasengan. So, I personally don't see him using Rasengan. I mean...Team Gai barely uses any ninjutsu. xD It's all taijutsu.
Jiraiya would probably note that Neji specializes in taijutsu, especially Hyuuga style, so he'd probably try to bring up and have Neji's taijutsu skills advance, afterall. So, the use of ninjutsu could be lessened to little use to no use at all.
Overall...Jiraiya and Neji? Jiraiya would be sooooo bored and Neji would be sooooo humiliated. xD Not by Jiraiya's power, but everything besides his power. And I don't think Jiraiya would really enjoy fooling around with a member of the Hyuuga clan. I don't think he really sees a reason to do so.
Also...
Hyuuga Neji + Rasengan =
rofl And no. It does not equal 'Unbeatable Neji'. It's still kinda short distance, don't you think? Something or anything like Sasuke's Ryuuka No Jutsu would take Neji out if he tried to use his Rasengan. Or Gokakyuu No Jutsu could do well too. No...Housenka No Jutsu! 'Cause that's alot of fireballs. >D That Rasengan would be gone by the time Neji is done taking them all out.
Not that I doubt Neji's skills. I wouldn't want to see Neji use Rasengan. His taijutsu skills right now are already off the hook. >D And Neji really needs to beat the s**t out of Sasuke for leaving. D< NO, wait, he should team up with Naruto and all the others that tried to get him back, and beat the living s**t out of that guy. >D
But I would want to see what Jiraiya would do to poor Neji. >) Neji would go nuts. Literally...and he might go perverted. xD And it might even reveal what other moves the oh great Jiraiya-sama has up his red sleeve. =D
...And finally after all that time wasted in consideration, put his Byakugan to good use, and hold it against Tenten...>D