The former New York mayor will be tasked with supporting UN Secretary-General Antonio Guterres' plan to host a major summit next year to take stock of progress in implementing the Paris climate agreement.
Trump last year stunned the world by announcing plans to pull the United States out of the Paris accord, signed by nearly 200 countries and parties. The US president is not expected to attend the 2019 summit.
"My hope is that President Trump listens to his advisers and looks at the data and changes his mind," Bloomberg told reporters as he met with Guterres to discuss his mission.
"If that's the case, that shows a great leader, who, when facts change and they recognize something different they are not bound to what they did before. They are willing to change.
"This president does change his views," added Bloomberg. "Generally, it's from one day to the next, but over a longer period of time, hopefully he will."
Bloomberg said he spoke to Trump once and that "he certainly knows my views on climate change", adding that his staff remained in touch with the US administration.
The UN envoy will be traveling to Cape Town this week as the South African city faces a crisis over its dwindling water supply due to drought and sparse rain.
Guterres praised Bloomberg as a "true leader" on climate which he described as "the defining question of our time".
Under the terms of the Paris agreement, the United States can formally give notice that it plans to withdraw in 2019, three years after the accord came into force.
The withdrawal would become effective a year later, in 2020.
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