H1B transfer initiated before I-94 expiry

I am on H1b with current employer A. H1B and I 94 is valid till Aug 31st 2021. Extension is already in progress. Now, I have a new employer (say B) who is willing to file for H1B transfer under premium. But the issue is my I-140 approval with current employer was approved on Feb 21st 2021 and it would be only after Aug 21st 2021, I can move to a new employer and still be able to retain the priority date. New employer would be filing for H1 transfer under premium in another 7 days. Can I join new employer after Aug 31st 2021 like in Sep 1st week, by then my current I-94 would have expired but H1B transfer is already initiated? I am worried about the possible few days gap between I94 expiry and H1B transfer approval with new employer.

There should be no issue as you have both, a timely filed extension of status and change of employer petition before the expiry of your current I94. You can start with new employer once your I140 complete 180 days.

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Thank you for the quick reply.