UPDATE 2-Peripheral euro zone bonds in demand as flash PMI dips

* Euro zone flash composite PMI dips

* Bond yields lower, reverse early rise

* Greek yields fall after comeback bond announced

* Euro zone periphery govt bond yields http://tmsnrt.rs/2ii2Bqr (Updates prices)

LONDON, July 24 (Reuters) - Bond yields in Italy and Portugal hit their lowest levels in around a month on Monday, leading a dip in euro zone borrowing costs as signs of slowing growth in business activity reaffirmed a view that any unwinding of ECB stimulus is likely to be slow.

In the past week, comments from the European Central Bank and the euro's strength - trading near two-year highs against the dollar - have eased investor worries about a tapering of the central bank's bond-buying stimulus scheme.

Data on Monday showing Germany's private sector grew at a slower pace in July, while French business activity slowed more than expected in July to a six-month low gave investors another incentive to move back into bond markets that have been rattled in the past month by concerns over tapering.

Portugal's 10-year government bond yield fell as much as 5 basis points to 2.89 percent, its lowest level in just over a month. Italian equivalents fell 4 bps to 2.03 percent - their lowest in almost a month.

"Sentiment is still very positive after the ECB sounded dovish at last week's meeting," said Benjamin Schroeder, a rates strategist at ING. "It doesn't look like there is an urgent need to tweak policy."

Greek bonds were also in high demand after a positive ratings review and the announcement of its imminent return to financial markets after three years in exile.

S&P on Friday revised the ratings outlook for Greece to "positive" from "stable", citing an expectation that debt levels will decline.

On Monday, Athens invited holders of its 4.75 percent outstanding bonds maturing in 2019 to tender them for cash, along with a plan to offer new five-year paper.

Most other euro zone bond yields were a touch lower on the day, with Germany's benchmark 10-year Bund yield dipping to its lowest level in just over a week of 0.49 percent having initially traded higher.

Analysts said the initial weakness in bond markets was caused by an upward revision in International Monetary Fund growth forecasts for the euro area. When a bond yield rises, its price falls.

In an updated World Economic Outlook on Monday, the IMF upgraded its 2017 gross domestic product (GDP) growth projection for the euro zone to 1.9 percent, up 0.2 percentage point from April, and pointed to "solid momentum".

For Reuters Live Markets blog on European and UK stock markets see reuters://realtime/verb=Open/url=http://emea1.apps.cp.extranet.thomsonreuters.bi z / c m s / ? p a g e I d = l i v e m a r k e t s

(Reporting by Dhara Ranasinghe; Editing by Adrian Croft)