You should be able to get transfer
Sathish,
There is nothing in law that suggests that you should be working for your original petitioner if you want to seek a transfer.
H1B cap is congressionally mandated to limit overall intake per year. Once you have qualified through the cap you can apply to any employer you wish and ask them to file for H1B transfer. USCIS documentation says following
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"This limitation, commonly referred to as the “H-1B cap,” does not apply to H-1B petitions filed on behalf of aliens who have been counted against the cap in the previous six years and who have not been outside of the United States for one year or longer. Thus, generally, petitions to extend an H-1B nonimmigrant’s period of stay, change the conditions of the H-1B nonimmigrant’s current employment, or request new H-1B employment filed on behalf of H-1B workers already in the United States will not count against the H-1B fiscal year cap"
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You are on L1 and contractors who sponsor H1 do not know intricacies involved here. Continue on your L1 job, but ask your new employer to file for a transfer and join them as and when transfer comes through ( they can even file for premium processing which is decided within 15 days). USCIS asks for Payslips to ascertain legal status and not for checking if you have already joined the petitioner. Contractors generally misinterpret this as USCIS expects the beneficiary to join the petitioner before seeking transfer. Form 129 for change of status mandates pay stubs and it has nothing to do with H1B. Hope this clarifies.
