UPDATE 3-Oil steady near $110 on supply risks, U.S. demand

* Ukraine government vows to continue offensive vs rebels

* U.S. crude inventories up, gasoline down - EIA

* Coming Up: Univ. of Michigan sentiment, outlook at 1355 GMT

(Updates throughout, changes dateline, previous SINGAPORE)

LONDON, May 30 (Reuters) - Brent crude oil steadied around $110 a barrel on Friday, near the top of its range over the last three months, underpinned by supply worries and evidence of strong oil demand in the United States, the world's top oil consumer.

Fighting continued in eastern Ukraine with the defense minister promising to push ahead with an offensive against rebels until "normal life" was restored.

Investors worry the Ukraine crisis will escalate tension between Russia, one of the world's largest oil and gas exporters, and the West, jeopardizing energy supplies on which Europe is heavily dependent.

Oil supply from Libya, a source of high quality light crude, remains minimal with protests and violence disrupting output.

U.S. oil inventory figures were also supportive.

Gasoline stocks fell 1.8 million barrels, data from the U.S. Energy Information Administration showed, compared with expectations of a 300,000-barrel gain, indicating a strong start to the U.S. summer driving season.

Brent crude was down 10 cents at $109.87 a barrel by 0800 GMT, but was up nearly 2 percent for the month.

U.S. light crude oil slipped 25 cents to $103.33, partly under pressure from news of an overall rise in U.S. crude stocks. U.S. crude futures are up nearly 4 percent this month.

"Potential downside for the oil price now seems quite limited," said Carsten Fritsch, senior oil and commodities analyst at Commerzbank in Frankfurt. "Risk, particularly Ukraine risk, is supporting the market."

Iran's crude oil exports increased in May after a decline in April, according to sources who track tanker movements. Iran's exports have averaged 1.38 million bpd in May, one of the sources said.

Asia's top buyers took in an average 1.24 million bpd of the Islamic republic's crude between January and April, up nearly a third from the same period last year.

The higher Iranian sales in late 2013 and early 2014 have led to some concerns in Washington that a softening of sanctions has given Tehran's economy a boost, but officials in the Obama administration have said they expect Iran's oil sales to meet the target for the entire six-month period.

"As long as the talks and the openness on the nuclear program continues to proceed, then it shouldn't be a big problem," said Richard Gorry, head of JBC Energy Asia.

(Additional reporting by Manash Goswami in Singapore; Editing by William Hardy)