The SpaceX and Tesla founder was initially an outspoken opponent of Mr Trump and his campaign but when the property magnate and reality TV star won the race to the White House he indicated an interest in working with him.
Mr Musk has was a guest at the new president’s initial "tech summit" at Trump tower and a number of other meetings since.
Now, in a series of tweets, he has praised Mr Trump’s new Secretary of State Rex Tilllerson's belief that climate change exists and that something should be done about it.
Elon Musk arrives at Trump Tower for talks with the new US President
Elon Musk and fellow businessmen at a meeting with Donald Trump
I’m just saying that we should see what happens first. The actions may be surprising
He welcomed Mr Tillerson’s support for a carbon tax and said it was "what is really needed to move the needle".
In response to a Twitter user who questioned whether Mr Tillerson, a former boss of oil and gas giant ExxonMobile, actually believed in climate change, Mr Musk said he should wait and see.
He tweeted: ”I’m just saying that we should see what happens first. The actions may be surprising."
US Secretary of State Rex Tillerson
Mr Tillerson proved to be a controversial selection by Mr Trump but after he was appointed to the foreign affairs job by a slim margin Mr Musk insisted he had the "potential" to be "excellent" in the job.
In a post directed at The Economist magazine, which had tweeted an editorial praising Mr Tillerson's "integrity", Mr Musk said he agreed.
Elon Musk initially opposed Donald Tump but now thinks he can work with him
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He posted: ”This may sound surprising coming from me, but I agree with The Economist.
"Rex Tillerson has the potential to be an excellent Sec of State."
In later updates, Mr Musk wrote: "Rex is an exceptionally competent executive, understands geopolitics and knows how to win for his team. His team is now the USA."