By Lisa Richwine and Marina Lopes

LOS ANGELES/NEW YORK (Reuters) - Verizon Communications Inc on Thursday demanded that Netflix Inc immediately stop displaying messages to customers that place blame on Verizon's broadband service for slow delivery of Netflix TV shows and movies.

The letter is the latest sign of tension between content providers like Netflix and Internet service providers over who should pay the price for companies that stream heavy traffic online.

In a cease-and-desist letter sent to Netflix, Verizon also asked the video streaming service to provide information including a list of customers on the Verizon network to whom Netflix delivered the notices, or face legal action.

"Failure to provide this information may lead us to pursue legal remedies," Verizon general counsel Randal Milch said in a letter to Netflix general counsel David Hyman.

In mid-May, Netflix started a test of messages displayed on the screen for some customers when a video is buffering. The messages say that there is congestion on the network of Verizon or another Internet service provider.

Netflix said on Thursday the test is continuing and meant to provide customers more information about their service, similar to a speed index that Netflix has published for months with a ranking of Internet service providers.

"This is about consumers not getting what they paid for from their broadband provider," Netflix spokesman Jonathan Friedland said. "We are trying to provide more transparency, just like we do with the ISP Speed Index, and Verizon is trying to shut down that discussion."

David Young, Verizon's vice president of federal regulatory affairs, told Reuters he felt the messages were part of a Netflix agenda to promote its views in Washington. Netflix has been urging the Federal Communications Commission to prevent broadband providers such as Verizon from charging Netflix fees for faster delivery of videos.

"That seems to be the motivation, to try to make a point or make points in that discussion that is going on in Washington D.C.,” Young said in a phone interview, though he wouldn't specify the issues.

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In the letter, Verizon said the message's assertions were deceptive and inaccurate and that the responsibility for slow streaming on the site falls on Netflix.

Verizon said that in order to save money, Netflix relies on middlemen to distribute its content knowing that some of them have issues with congestion in some networks.

In April, Netflix said it reluctantly signed a deal to pay fees to Verizon to bypass those middlemen and deliver content directly to the company, ensuring faster speeds. But Verizon is still working to implement the needed architecture and expects to finish improvements by the end of 2014, said Young.

"I sincerely hope this is not a harbinger of things to come in terms of how Netflix treats its partners and our mutual customers," Milch said in the letter.

The Verizon notices came to light this week when a reporter who was a Verizon user tweeted about them. Comcast Corp, which also receives connection fees from Netflix, is not aware of any reports of Netflix buffering messages to its customers, a Comcast spokeswoman said.

(Reporting by Lisa Richwine and Marina Lopes; Additional reporting by Ron Grover and Alina Selyukh; Editing by David Gregorio and Jonathan Oatis)