The president approvingly quoted an accusation by billionaire investor and Trump supporter Peter Thiel that the company had been "working with the Chinese government".
He said that Mr Thiel was "a great and brilliant guy who knows this subject better than anyone!"
He then wrote: "The Trump Administration will take a look!"
Mr Thiel – who helped found PayPal and serves on Facebook's board – had suggested that Google had been infiltrated by Chinese engineers and had made the "seemingly treasonous decision to work with the Chinese military", according to a report from Axios over the weekend.
Speaking at a conference focused on nationalism, Mr Thiel said that three questions need to be asked of Google, according to the same Axios report. He said that the company should be asked how many foreign intelligence agencies had infiltrated its work on AI, whether the company's senior management thinks it has "been thoroughly infiltrated by Chinese intelligence", and why it was working "with the Chinese military and not with the US military".
He suggested those questions should be asked "by the FBI, by the CIA, and I'm not sure quite how to put this, I would like them to be asked in a not excessively gentle manner."
Mr Trump appears to have seen reports of Mr Thiel's remarks on Fox & Friends, the news show he is widely reported to watch faithfully each morning.
He gave no details on what form any investigation could take, who would lead it, or how it would actually be launched. He also gave no indication of whether he had any information to show that Mr Thiel's claims are true.
There are no public reports that Google has been infiltrated by Chinese intelligence.
In the same speech, Mr Thiel had railed against other problems he claimed blight US society. He said that Americans were too fat and addicted to drugs, but that "the biggest problem in our society is the problem of political correctness".
Mr Trump's message came amid his continued defence of racist remarks in which he attacked four Congresswomen of colour. He continued the attacks on those politicians in posts either side of the tweet about Google.