The Israeli Air Force Boeing 707 was manoeuvring during a refuelling stop when the accident happened late on Thursday night.
An airport spokeswoman said the truck was being operated by handling agents Servisair.
Its driver was taken to hospital as a precaution but later allowed home.
There were no casualties on board the aircraft.
Mr Barak's senior policy adviser, Danny Yatom said: "It is good that it happened on the ground and not in the air."
Missed conference
Mr Barak was due to fly on to Atlanta to make the keynote speech at the annual conference of Jewish communities in the United States.
He was understood to have been asleep at the time of the collision, during what should have been a one-hour refuelling stop.
The prime minister was travelling with an entourage of around 50 people on the jet, which is always used to carry him on overseas trips.
It later emerged that he spent the night at the Hilton Hotel at the airport and was making arrangements to film his contribution to the conference by satellite link.
Israeli Embassy press secretary David Schneeweiss said: "No-one was hurt but the plane was ruled unfit for travel. We have been working overnight to arrange new flights to the USA."
Mr Barak was planning to take a scheduled flight from Manchester to New York, where he is due to carry out a series of official engagements over the next few days.