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* Bund yields give up early falls as Deutsche shares recover
* Still on track for biggest monthly fall since June
* Irish 10-year bond yields fall to record low
* Portugal yields set for biggest monthly rise since June 2015 (Recasts with late-day move in yields, adds Portugal)
By Dhara Ranasinghe and John Geddie
LONDON, Sept 30 (Reuters) - Safe-haven German bond yields gave up their falls on Friday as Deutsche Bank shares bounced back from record lows, but remained on track to end September with their biggest monthly falls since June.
Deutsche, Germany's biggest lender, has been engulfed by crisis after being handed a demand for up to $14 billion earlier in September from the U.S. authorities for misselling mortgage-backed securities.
As concerns about the bank's health resurfaced this week, yields on Bunds - seen as a safe haven by investors - fell to their lowest levels since mid-August and opened lower on Friday after reports that a number of hedge funds had withdrawn cash from the bank, sending its shares to record lows.
Deutsche's chief executive sought to reassure staff about the future of the bank on Friday.
The bank's shares meanwhile surged about 6 percent just before the market close after a media report said an agreement with U.S. authorities over the alleged misselling of mortgage-backed securities was being discussed.
That in turn led to a turnaround in bond markets, where yields had spent much of the day lower.
"The move this morning was clearly Deutsche Bank based and since then we've had a rethinking of those fears," said Sergio Capaldi, fixed income strategist at Intesa SanPaolo.
By late trade, German 10-year bond yields were at minus 0.12 percent, flat on the day. They had fallen 4 basis points earlier to minus 0.161 percent, matching a level struck on Tuesday, which was the lowest since mid-August.
Still, Bund yields were on track to end September with a fall of almost 6 bps, the biggest monthly fall since June when Britain's unexpected referendum result in favour of leaving the European Union roiled global markets and saw a flight-to-safety.
"Valuations would justify higher yields, around 0 percent say, but in the current environment there is a flight-to-safety premium that is supporting Bunds," said Orlando Green, European fixed income strategist at Credit Agricole.
Other euro zone bond yields also gave up earlier falls which had pulled Ireland's 10-year bond yield to a record low of 0.312 percent.
With markets focused on Deutsche Bank there was little immediate reaction to news that euro zone consumer prices grew 0.4 percent in September, twice as fast as in August.
That still puts inflation well short of the ECB's target of just under 2 percent but might be enough to prompt questions about whether the ECB's ultra-loose monetary policy is, at last, starting to work.
For Reuters new Live Markets blog on European and UK stock markets see reuters://realtime/verb=Open/url=http://emea1.apps.cp.extranet.thomsonreuters.biz/cms/?pageId=livemarkets
On the ratings front, Spain is set to be reviewed by Standard & Poor's later on Friday, at the end of a week which has seen ructions in its Socialist Party renew concerns about political stability in the euro zone's fourth-biggest economy.
Strategists at Commerzbank said the firm may change the stable outlook on the country's BBB+ rating to negative.
Spain's 10-year bond yields fell 4 basis points to 0.88 percent.
Elsewhere, Portugal's 10-year bond yield was set to end September with a rise of about 28 bps.
That would mark its biggest monthly increase since June last year, reflecting growing concerns about the country's ratings outlook. (Editing by Janet Lawrence)