Can H1B Transfer & PERM applied in 6th year ending in 2 months

I’m Gouthami from California US. I would like you to assist me with my Visa query. Currently, I have 60 days left before my H1B expires, I do not have any PERM / I140 applied. I have been a full-time employee all 6 years with No job change

I have been working in the US on an H1B Visa since March 17, 2015, including two H1B extensions. My current extension allows me to remain in the US until mid-July, 2021. Currently, I work for a Client company through a consulting firm. My Client has just informed me that they are very interested in hiring me directly, at which point I would want to get the ball rolling towards obtaining a green card. Considering that I am in my sixth year, and this would involve transferring my H1B to a Client from my existing employer.
If the Client hires me, what are my best options, and what is the appropriate sequence of steps to expedite moving from my current H1B status through the PERM and I140 process to obtaining my green card? Are there any options where if the Client has at least filed applications on my behalf, that I could remain in the US while the application is in the process? If not what do the sequence and timing look like?
Please advise what would be the options for me?

Thank you in advance for your advice on how to handle this matter.

Regards,
Gouthami

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