Extend J1 while applying for H1B

[chlzr]

Hi all,

Right now I'm on a J1 visa for an internship that expires on April 29th, but it can be extend until January 20th.
The employer offered a permanent position, and is going to file for an H1B visa. So my question is as follows: Is it possible to apply for an extension of the J1 visa until (let's say) 30th September while having applied for the H1B?

Thanks!