The property tycoon hailed Putin as a man "highly respected within his own country and beyond". The remarks came after the Russian leader said Trump was a "very colourful, talented person" during his annual news conference.
The two men - both known for their blunt manner of speaking - do not know each other personally. After hearing Putin's comments on Thursday, Trump released a statement praising the Russian leader.
"It is always a great honour to be so nicely complimented by a man so highly respected within his own country and beyond," he said.
"I have always felt that Russia and the United States should be able to work well with each other towards defeating terrorism and restoring world peace, not to mention trade and all of the other benefits derived from mutual respect," he added.