* Stocks bounce pushes most euro zone bond yields up
* Recovery in risk appetite benefits Italian market
* IMF says Italy needs to respect EU budget rules
* Moody's to review Portugal ratings
* Euro zone periphery govt bond yields http://tmsnrt.rs/2ii2Bqr (Updates with Draghi comments,)
LONDON, Oct 12 (Reuters) - German government bonds were set for their best week since August on Friday thanks to the equities rout, with dovish comments on the inflation outlook from ECB chief Mario Draghi supporting broader euro zone debt markets.
Draghi said on Friday that underlying inflation in the bloc will rise gradually, toning down earlier remarks which foreshadowed a "relatively vigorous" rise.
Having risen in early trade in response to a firmer tone in world stock markets, most yields in the euro zone crept back down to stand little changed on the day.
In Germany, the bloc's benchmark bond issuer, 10-year yields were marginally higher at 0.52 percent and below 4-1/2-month highs reached earlier this week at 0.58 percent.
They are down about 4 bps this week, the first decline in six weeks and the biggest drop in two months.
While world stocks recovered some ground on Friday after this week's rout, sentiment remained fragile. European shares were last up 0.5 percent and off their highs.
This biggest stock market shakeout since February has been blamed on a series of factors, including worries about the impact of a Sino-U.S. trade war, a spike in U.S. bond yields and caution before the earnings season.
"We're still left with the sense that there has been a significant shift that markets now have to take stock of," said Chris Scicluna, head of economic research at Daiwa Capital Markets in London.
For some analysts, the relatively small fall in German bond yields was a sign of investor reluctance to buy fixed income.
"Ten-year German yields failed to close through 0.50 percent in spite of the Italian budget turmoil at the start of the week, and despite the global stocks meltdown," Mizuho analysts said in a note.
Italy's bonds stabilised , with yields edging 2-3 bps lower as risk appetite recovered. Analysts said a lot of negative news has now been priced into the market.
Italian 10-year bond yields rose to 4 1/2-year highs earlier this week on tension between Rome and the European Union over the government's expansionary budget plans.
The International Monetary Fund said on Friday Italy needs to respect EU budget rules and build a cash buffer for the next economic downturn.
Portugal was also in focus amid expectations that ratings agency Moody's would lift its rating by one notch later on Friday.
"It's our base case that Portugal will be upgraded to investment grade, it is overdue," said Michael Leister, rates strategist at Commerzbank.
Sunday's election in the German state of Bavaria was also in focus. Chancellor Angela Merkel's Bavarian allies are heading for their worst showing in a state election in over 60 years, a setback that risks widening divisions within her crisis-prone national government.
(Reporting by Dhara Ranasinghe, editing by Larry King and David Stamp)