* Dollar index struggles near 8-month trough
* Euro holds firm, not far from this year's peak of $1.3711
* Yen near 5-week high vs dollar, little changed after BOJ
By Anooja Debnath
LONDON, Oct 4 (Reuters) - The dollar struggled near an eight month low on Friday as the U.S. government shutdown dragged on, while the euro, boosted by positive euro zone data, held firm within striking distance of its 2013 peak.
The dollar was flat at 79.767 against a basket of currencies, not far from the previous session's trough of 79.627, its lowest since early February. The dollar was on track for its fourth consecutive week of losses.
The euro was flat at $1.3621, having touched a peak of $1.36465 on Thursday, its highest since February when it scaled this year's high of $1.3711. It has risen 0.8 percent on the dollar so far this week.
The dollar saw little respite this week with markets concerned that the U.S. impasse would merge with a more complex fight over raising the U.S. debt limit later this month. Failure to do so may lead to a historic debt default.
Adding to the greenback's woes was data on Thursday that showed growth in the U.S. service sector had cooled last month.
"No one wants to touch the dollar while we have uncertainties regarding the U.S. government shutdown. We also had a disappointing service sector number and that also added to the negative dollar sentiment," said Niels Christensen, FX strategist at Nordea.
Adding to market uncertainty, the U.S. Labour Department on Thursday said the employment report for September will not be released as scheduled on Friday due to the government shutdown. No new date was set.
Thus, any confirmation of an improving labour market that the Federal Reserve wants to see before cutting its stimulus, will likely be delayed and hurt the dollar. Two senior Fed officials said monetary policy was being kept easier to help offset the harm caused by political fighting.
"Now some people are saying (Fed) tapering will have to wait until next year," said Katsunori Kitakura, associate manager of market making at Sumitomo Mitsui Trust Bank. "Those who have been expecting it in October should be having a bit of panic now. Those who have bet on December may be worried too."
The dollar's weakness helped support the euro.
The resolution of political turmoil in Italy, the European Central Bank sounding less perturbed about the euro's strength and refraining from any immediate policy action to help the economy and positive service sector data from the currency bloc, all supported the euro.
Against the struggling dollar, the yen traded near a five-week high, reacting little to the Bank of Japan's decision to keep policy on hold as it was expected.
The dollar was down 0.1 percent at 97.12 yen, having slid to a five-week low of 96.93 yen on Thursday, not far from an important support of its 200-day moving average at 96.63.