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UPDATE 3-Amazon finishes higher despite Trump's new threat on shipping rates

* Shares of Amazon close up after day of ups and downs

* President renews criticism over retailer's use of USPS

* Trump says Amazon does not pay fair shipping rate

* White House not considering taking action -report (Recasts with share volatility and report that White House not preparing any action; updates share action; adds background)

WASHINGTON, April 3 (Reuters) - Amazon.com Inc's shares took another ride on the roller-coaster on Tuesday, falling as President Donald Trump repeated his unsubstantiated claim that deliveries for the world's biggest online retailer cost the U.S. Postal Service money, then rising on a report that the White House is not considering any action against Amazon.

The value of Amazon shares held by Jeff Bezos, the online retailer's chief executive and single largest shareholder, had taken a $10 billion hit in the week since Trump began attacking him and his company on Twitter.

Citing an unnamed report, Trump told reporters at the White House the company is not paying the USPS a fair rate, costing U.S. taxpayers billions of dollars and forcing other retailers out of business, and he threatened to raise rates.

It was the latest salvo in a string of attacks as Trump stepped up his criticism of Amazon and Bezos, who also privately owns The Washington Post, one of Trump's favorite punching bags in his claims over "fake news."

Bezos, whom Forbes magazine ranks as the world's richest man with an estimated net worth of $115.6 billion, owns 78.89 million Amazon shares, worth about $110 billion at Tuesday's market close.

Amazon shares closed up 1.5 percent at $1,392.05. The shares had started the day higher but fell as low as $1,355.33 after Trump's latest Amazon-related tweet.

On Monday Trump had also attacked the company over its shipping after criticizing it last week over taxes.

"The post office is losing billions of dollars ... because it delivers packages for Amazon at a very low rate," Trump told reporters on Tuesday. "If you look at the cost that we're subsidizing, we're giving a subsidy to Amazon."

Trump offered no details about the report he cited or how he planned to charge the company more through USPS.

But inside the White House there are no active discussions about taking action against Amazon, Bloomberg later reported, driving Amazon's shares back up.

White House officials said the same thing last week after Trump attacked Amazon in a tweet.

Amazon also ships packages through providers such as FedEx Corp and United Parcel Service Inc as well as its own experimental shipping service. Representatives of Amazon and USPS had no comment on Trump's tweet on Tuesday and could not be immediately reached regarding his latest comments to reporters.

The business community, however, criticized Trump.

Its inappropriate for government officials to use their position to attack an American company," said Neil Bradley, chief policy officer of the U.S. Chamber of Commerce, citing the value of the free enterprise system and the rule of law. "The record is clear: deviating from those processes undermines economic growth and job creation. (Reporting by Steve Holland Additional reporting by Makini Brice and Lisa Lambert Writing by Susan Heavey Editing by Bill Rigby and Leslie Adler)