* Asian manufacturing data spurs global growth concerns * Investors mull last week's European rescue proposals * US manufacturing growth seen slowing in June By Chris Reese NEW YORK, July 2 (Reuters) - U.S. Treasury debt prices rose on Monday due to safe-haven buying as weak Asian data spurred worries over the pace of global growth, although gains were tempered after last week's agreement to give euro zone rescue funds more flexibility. Purchasing managers surveys out of China, Japan, South Korea and Taiwan showed demand from importing centers such as Europe and the United States slowed in June. "Treasuries are modestly higher after weaker Chinese PMI data and record high eurozone unemployment readings," said William O'Donnell, head of U.S. Treasury strategy at RBS Securities in Stamford, Connecticut. That took the shine off an agreement by European leaders to let their rescue fund inject aid directly into stricken banks from next year and intervene in bond markets to support troubled members. Having had the weekend to digest the new measures, some investors are now questioning whether the rescue fund will have enough firepower to cool down any selling pressure in the large Italian and Spanish debt markets. Worries are also growing over a long implementation process. Benchmark 10-year Treasury notes were trading 4/32 higher in price to yield 1.63 percent, down from 1.64 percent late Friday. "Treasury 10-year yields are opening up today's session in the middle of the same, mind-dulling 1.57 percent to 1.67 percent opening-closing range that we've seen in 18 out of the last 19 sessions," O'Donnell said. Investors are looking for some possible market guidance in data on U.S. manufacturing at 10 a.m. EDT (1400 GMT). The median of forecasts from analysts polled by Reuters is for the Institute for Supply Management's index of national factory activity to have slipped to 52 in June from 53.5 in May. Ahead of the data, 30-year bonds were trading 7/32 higher in price to yield 2.74 percent, down from 2.75 percent late Friday.