According to a Vox report, Bezos called the fellow billionaire in February this year and asked him to consider entering the 2020 US presidential race.
"Bloomberg told Bezos no at the time, according to a person briefed on the phone conversation," said the report.
Bloomberg also had a question for Bezos: Will he reconsider his decision to cancel plans for an Amazon headquarters in New York City?
Bezos replied saying he would not reconsider his decision not to open a second Amazon headquarters in New York.
Amazon in February had announced to abandon its project to build a mega campus in Long Island City, Queens after widespread protests.
The Democrat US presidential candidate Warren on Saturday mocked both the billionaires.
"One billionaire calls another billionaire and asks him to run for president I'm shocked! Here's the deal: Companies like Amazon have too much power, and billionaires like @JeffBezos and @MikeBloomberg should pitch in so that everyone can succeed," Warren tweeted.
According to reports, Bloomberg, 77, is reportedly preparing to enter the Democratic race for the 2020 US presidential election.
Bloomberg, who co-founded financial information and media company Bloomberg LP in 1981, was mayor of New York City from 2002 to 2013, serving three terms.
Bloomberg's net worth is $52 billion, according to Forbes magazine, which means he's one of the world's wealthiest individuals.
( With inputs from IANS )