The activists also damaged greens and sprayed blood-red paint over the luxury resort's clubhouse overnight on Saturday.
Trump sparked outrage last month when he suggested the US “take over” Gaza and turn it into the “Riviera of the Middle East”, while forcing its Palestinian inhabitants to relocate to Egypt or Jordan.
One scene showed Trump and Israeli Prime Minister Benjamin Netanyahu sipping cocktails in swimsuits by a pool.
Police Scotland told AFP it was investigating after receiving a report of damage at around 0440 GMT on Saturday.
The Turnberry property in southwest Scotland is one of two resorts that Trump owns in the country, his mother's ancestral home.
“Turnberry is a national treasure and will continue to be the number one beacon of luxury and excellence in the world of golf.”
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