The State Department originally notified Congress of the sale last September during President Barack Obama's administration. It was pulled back because of concerns that Bahrain had not made promised improvements to its human rights record.
The Trump administration has separated the human rights issues from the transfer, the source said.
Members of Congress were not immediately available to comment on whether they would object to the sale over human rights concerns this time.
The $4.867 billion sale includes the aircraft, 23 engines, radars and other avionics, air-to-air and air-to-ground ordnance and related equipment.
The latest notice sent to Congress provides for 40 days of additional congressional review, then a formal notice to Congress as required by the Arms Export Control Act, after which the licenses for the sale would be approved.