* EU's Moscovici says wants to reduce tension with Italy
* Italian bond yields fall across the curve
* BTP/Bund spread falls from 5-1/2 year highs (Updates with detail on Moscovici comments, price action)
By Dhara Ranasinghe and Virginia Furness
LONDON, Oct 19 (Reuters) - Italy's government bond yields fell from 4-1/2 year highs and its stock market rallied on Friday after European Economic Affairs Commissioner Pierre Moscovici said he wanted to reduce tensions with Italy over the country's contentious budget plans.
Speaking at a news conference after a two-day visit to Rome, Moscovici said Brussels shared Italy's declared goals of boosting growth and cutting debt, and reiterated that no decision had yet been taken over the budget.
He said he did not fear contagion to other euro zone countries stemming from the market tensions over Italy.
His remarks brought relief to Italian markets just a day after a letter from the European Commission stepped up pressure over the budget, labelling it an unprecedented breach of EU fiscal rules, and sparking another violent selloff.
"It is hard to believe that he (Moscovici) is making these comments having sent the letter yesterday. It shows his change of tone and indicates that Brussels is mindful of the market reaction and the corresponding risks," said Michael Leister, rates strategist at Commerzbank.
"There was a new dimension in the sell-off in reaction to this letter, Spain and Portugal showed clear sings of contagion and Brussels has realised it wasn't the best strategy to escalate the situation."
Italian bond yields were up to 17 basis points lower on the day, fully reversing earlier rises.
Ten-year bond yields were down 10 bps at 3.57 percent , down from a 4-1/2 year high hit earlier at 3.78 percent.
Thirty-year bond yields fell from a 4-1/2 year high of around 4.15 percent, while two-year yields were down 17 bps after rising almost 20 bps on Thursday.
The 10-year bond yield gap over safer Germany -- effectively the risk premium on Italian bonds -- narrowed to 312 bps, having hit its widest in 5-1/2 years at around 338 bps.
As Italy's bond market rallied, the country's top stock index climbed into positive territory, trading up 0.25 percent. The bank stocks index also rapidly pared losses, last trading up 0.03 percent.
Banking stocks had come under heavy selling pressure earlier in the day from the renewed spike in sovereign bond yields.
Italian lenders hold around 375 billion euros of domestic bonds - accounting for 10 percent of their assets - and the spike in yields, by eroding the value of those holdings, eats into their capital levels.
Earlier on Friday, five-year credit default swaps in Italian bank UniCredit rose to 204 basis points, from a closing price of 194 bps on Thursday and was the highest since Dec 2016.
"Ultimately there could be a good outcome but we will get a lot of negative headlines in between that," said Timothy Graf, head of EMEA macro strategy at State Street.
"But Italy isn't Greece. Its external balances are healthier and it's a much more important economy and I don't think the brinkmanship that was allowable with Greece is allowable with Italy," he said, referring to the Greek debt crisis.
The recovery in Italian debt also weakened demand for bond safe havens.
Germany's 10-year Bund yield rose 2 bps at 0.44 percent , having hit its lowest in almost six weeks at 0.39 percent. Two-year German bond yields rose from two-month lows at minus around 0.67 percent.
Reporting by Dhara Ranasinghe and Virginia Furness; Additional reporting by Helen Reid; Editing by Martyn Herman