The recommended qualifications specifically indicate turboprop or turbine PIC however if you know Fred or a couple Senators there may be some leeway, but aside from that you'll need the kerosene burners. Note it says "typical qualifications"

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Typical qualifications for consideration as a FedEx pilot.

• Commercial Pilot Certificate with Multi-engine and Instrument rating (without limitations)
• Current ATP Certificate or written
• Current FE Turbojet Rating or written (FEX or Basic/turbojet)
• Must pass FAA mandated drug screen
• Recency and type of experience is considered
• Meet requirements for and currently hold First Class Medical certification
• 20/20 correctable vision
• College Degree from an accredited college or university
• Eligibility for rapid visa issuance, issued by offices in the United States to fly to any FedEx destination.
• Ability to obtain clearance from United States Postal Service for handling or access to U.S. mail, which includes FBI fingerprint check, and candidate must have resided in the United States for the last five consecutive years (except for U.S. military assignments)
• Eligibility for issuance of US Security Clearance
• 1500 hours total fixed-wing time as pilot-in-command (PIC) or second-in-command in multi-engine turbo-prop A/C or jet A/C or combination thereof, including a minimum of 1000 hours total fixed-wing pilot-in-command in multi-engine turbo prop A/C or jet A/C or combination thereof.
  Note: PIC for this purpose is defined as Captain/Aircraft Commander of record, not simply the sole manipulator of the controls.
  Note: FedEx considers only pilot time in fixed wing aircraft toward minimum qualifications. This does not include simulator, helicopter, flight engineer, bombardier, navigator, RIO, EWO, WSO, NFO, or Special Crew.
• All certificates and ratings required to be U.S.A. FAA issued