* Downbeat data add to Fed, ECB easing expectations * Activity thin ahead of Fourth of July holiday in U.S. * Aussie underpinned after RBA leaves rates on hold By Jessica Mortimer LONDON, July 3 (Reuters) - The euro was steady against the dollar on Tuesday as prospects of weakness in the U.S. economy leading to further monetary easing offset poor euro zone data and doubts about a European plan to support indebted countries. Analysts and traders said the euro could make some gains as investors trim bearish positions built in the run-up to last week's European summit, though a likely rate cut by the European Central Bank would limit gains. Data showing U.S. manufacturing activity contracted for the first time in nearly three years increased the chance of more easing by the Federal Reserve, which is likely to weigh on the dollar and bolster growth-linked currencies. But signs out of the euro zone were also discouraging, keeping sentiment towards the euro bearish. The jobless rate rose to a record in May and factory activity contracted again in June. The euro was steady at $1.2583, below a peak of $1.2693 hit on Friday in the wake of the summit agreement but comfortably above last week's trough of $1.2405. Traders cited a large option expiry at $1.26. "There are a lot of implementation risks but for now the market seems to be buying what came out of the EU summit. Before we go back to a trend of selling the euro we may have a bit of consolidation and perhaps some short covering," said Carl Hammer, chief currency strategist at SEB. He said investors would still "chase the euro lower" but against currencies other than the dollar, particularly Scandinavian and commodity-linked ones such as the Australian and New Zealand dollars. The dollar was steady against a basket of currencies, with the dollar index at 81.890, above Monday's low of 81.534. Against the safe-haven yen the euro was up 0.4 percent at 100.40 yen, still well below post-summit highs above 101.00 yen. Activity was relatively thin, however, ahead of the Fourth of July U.S. holiday, traders said. ECB AWAITED Many traders expect the ECB to move on Thursday to bolster the euro zone economy by cutting its main refinancing rate by 25 basis points to 0.75 percent. Jaco Rouw, fund manager at ING Investment Management, said there were differing opinions as to the impact of any rate cut, but he expected the euro to weaken given the poor economic outlook for the currency bloc. "One opinion is that the rate cut might boost risk sentiment and reduce the risk premium in the euro which could be positive, but a lower rate in itself would be euro negative," he said. "In the longer term, based on economic developments, there is still room for more monetary easing in Europe so we would position for a weaker euro." Some also expect the Federal Reserve to announce a third round of asset purchases, dubbed QE3, perhaps as soon as the U.S. central bank's next policy meeting from July 31 to Aug. 1. Australia's central bank held its main cash rate steady at 3.5 percent on Tuesday, to continue to gauge the effect of back-to-back cuts. The Australian dollar was up 0.1 percent at $1.0258, close to an earlier two-month high of $1.0285. The U.S. dollar bought 79.70 yen, down from Monday's high around 79.98 yen. Traders said dollar buying by model funds was offset by selling by Asian retail accounts. The dollar faced chart resistance at the top of its daily Ichimoku cloud, a key technical indicator, around 80 yen.