April 24 (Reuters) - ICE cotton futures fell on Monday, snapping a three-session gain on the first notice day for the May contract, on weaker demand for the natural fiber.

The July cotton contract on ICE Futures U.S. settled down 0.68 percent at 78.79 cents per lb, after touching the highest since March 20 at 79.63 cents earlier in the session.

"The July contract has reversed. It's showing weakness and that's a little confusing because the sales have been so good. We hear demand is softening, it's not nearly as good as it was two or three weeks ago," said Rogers Varner, president of Varner Brokerage in Cleveland, Mississippi.

The May cotton contract on ICE Futures U.S. settled up 1.39 percent at 80.09 cents per lb on Monday. "The May contract has liquidated... There were a couple of receivers, in the small amount of notices tendered. May is therefore history," Rogers said.

* The July cotton contract on ICE Futures U.S. settled down 0.54 cent, or 0.68 percent, at 78.79 cents per lb. It traded within a range of 77.78 and 79.63 cents a lb.
* Total futures market volume fell by 1,161 to 25,884 lots.
* Data showed total open interest gained 2,668 to 242,874 contracts in the previous session.

(Reporting by Karen Rodrigues in Bengaluru; Editing by Sandra Maler)