If you stay w/ this employer for at least 1 pay period, then pay checks will be needed by new employer’s petition.
If you are unable to provide the pay checks then there is a good chance that USCIS will approve the petition w/o I-94 attached. So you have to leave US and re-enter on new employer’s petition in order to work for them.
If you stay w/ current employer for less than 1 pay period or are getting all pay checks from them, then I don’t see any issue why they shouldn’t approve H-1 w/ I-94 attached.