Is it possible to rent a motorcycle near Manila airport?
The answer is yes. You can rent a motorcycle using the book2wheel.com website and ask the owner to deliver the motorcycle to the airport. The Owner will charge you a one-way delivery fee of 500 pesos or 300 pesos. Depending on how close his location is to the airport.
If someone offers you a motorcycle to deliver to the airport for a 500 delivery fee you should take it. The taxi driver will take you the same and waste your time. Most taxi drivers in Manila will try to get more money from you and will not return the change.
Here are some of the motorcycles and cars which you can rent near Manila Airport

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Honda PCX 150cc is the perfect motorcycle for long rides. You can rent it for 800 pesos. Brand new unit located in the Parañaque area which is close to the airport.

You can rent SUVs in the Makati area for just 1800 pesos. You can book directly at book2wheel.com. Rent a car in Manila if you are coming with your family. It will be way cheaper compared to renting a taxi.