A rival of the world's biggest ad agency doesn't think Amazon is already a big competitor to Google in online advertising

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Omnicom CEO John Wren.

Omnicom CEO John Wren downplayed the size of Amazon's advertising business in his company's first quarter earnings call.

This is in contrast to the thoughts of WPP CEO Sir Martin Sorrell, who is already preparing for Amazon's push into advertising and said the online retailer was what kept him up at night.

During WPP's full year earnings call in March, Sorrell said: "The answer to the question, 'What worries you when you go to bed at night and wake up in the morning?' isn't a 3-month-old child. It's Amazon - which is a child but not three months." He also said WPP had already set up an agency in Seattle, Amazon's headquarters, to deal with the online retailer.

Wren said Google's ad business would continue to dominate in the near term, especially in search advertising. A large chunk of Amazon's advertising business comes from sponsored product ads, which places it head to head against Google's search ads.

"It's an important alternative and I would never underestimate over the longer run, what Amazon is capable of doing. If you are looking at 2017 or the more immediate future I'd only list it as an important alternative to Google, and that's who I anticipate is going to take most of the market share in the short run," Wren said on the earnings call.

According to investment bank Morgan Stanley, Amazon's advertising business was forecast to grow by 37% between 2016 and 2018 to reach $5 billion in revenue. This still puts it significantly behind Google's search advertising business, which generated $28 billion in revenue in 2017 just in the US, according to an eMarketer report.

Omnicom reported revenue growth of 2.5% to $3.59 billion in the first quarter of 2017. Wren remained cautious about the effects political changes would have on his business in 2017, though:

"While our revenue growth exceeded our internal targets for the quarter, we remained cautious as numerous geopolitical and macroeconomic events remain unresolved," Wren said in reference to upcoming elections in France, Germany, and the UK, the legislative changes in the United States, and the situation in Syria and North Korea.

The agency holding group has been getting a lot of attention from new client gains. Its newly created media agency Hearts & Science won two of the biggest brands, AT&T and Procter & Gamble, as clients and it beat WPP in a pitch for Volkswagen's media buying budget.