June 6 (Reuters) -ICE cotton futures firmed on Tuesday, drawing from upbeat sentiment in the grains market, though a stronger dollar kept the upside momentum in check.

* The front-month July cotton contract CTc1 rose 0.59 cent, or 0.7%, to 85.38 cents per lb, by 11:30 a.m. EDT (1530 GMT). It traded between 84.48 cents and 85.71 cents.

* "The grain market is progressing well today, which I am sure is helping cotton a little bit," said Jack Scoville, vice president at Chicago-based Price Futures Group.

* Chicago wheat climbed over 3% to a near three-week high as the breaching of a dam in southern Ukraine heightened worries about escalation in the war between major grain exporters Ukraine and Russia. GRA/

* Limiting gains, the dollar index .DXY rose 0.2%, making the natural fiber less appealing to other currency holders. USD/

* The U.S. Department of Agriculture (USDA) in a weekly crop report on Monday showed 51% of U.S. cotton was in good to excellent condition in the week ended June 4 versus 48% a week ago.

* "Weather conditions for the new crop are generally pretty good, but the old crop seems to be enamored with the lack of availability of cotton right now, and it should be deliverable," Scoville added.

* Elsewhere, Australia's summer crop production, which includes sorghum and cotton, is estimated to fall 8% to 5.1 million tonnes over the same period, but remain well above the decade average.

* In key producer India, monsoon onset was delayed by another two-three days, weather officials said on Monday. The monsoon's late start could delay the planting of rice, cotton, corn, soybean and sugarcane.

Reporting by Ashitha Shivaprasad and Ananya Bajpai in Bengaluru; Editing by Shilpi Majumdar