UPDATE 4-Brent steadies above $107 on Ukraine; rise in US oil stocks weighs

* Russia seeks to reassure West, Ukraine over troops

U.S. crude stocks rise 7.1 mln bbls, six-fold more than forecast - API

* Libya not in full control of re-opened eastern port

(Updates prices)

By Peg Mackey

LONDON, April 9 (Reuters) - Brent futures steadied above $107 a barrel as tension escalated over Ukraine, but a steep rise in crude oil stockpiles in the United States weighed on prices.

While the Ukraine crisis may not directly impact global oil supplies and trade, the risk premium on oil is rising as investors worry the Kremlin's standoff with the West could quickly take a turn for the worse.

Russia dismissed as "groundless" concerns in Kiev and the West over the presence of its troops near the border with Ukraine, saying they posed no threat, and accused Washington of fuelling tension in the region.

U.S. Secretary of State John Kerry on Tuesday said Russian agents and special forces were stirring separatist unrest in eastern Ukraine.

"The geopolitical concerns surrounding Ukraine which were heating up the market at the start of March appear to have made something of a comeback, while few would be surprised to learn that the handover of Libya's key crude export facilities is turning out to be anything but straightforward," said JBC Energy.

Libya's oil protection force is not in full control of the Zueitina port even after a deal with rebels to end the blockade of eastern oil terminals, as some militiamen are still at the facility.

The port, along with the country's two largest, Es Sider and Ras Lanuf, has been under the control of an eastern federalist group led by former guard member Ibrahim al-Jathran, who recruited men from within his ranks.

Brent crude rose 5 cents to $107.72 a barrel by 0945 GMT. U.S. oil, under pressure from the inventory data, slid 16 cents at $102.40.

U.S. crude inventories rose by 7.1 million barrels in the week to April 4, data from industry group the American Petroleum Institute showed, compared with analysts' expectations for an increase of 1.3 million barrels.

But a steeper-than-expected fall in gasoline stocks is setting a floor under the U.S. oil benchmark as the drawdown adds to evidence of a broad economic recovery and robust demand as summer driving season gets underway.

U.S. gasoline stocks fell by 3.6 million barrels, compared with expectations of a 700,000-barrel decline.

U.S. crude inventories fell by 2.4 million barrels in the previous week ended March 28, as shown by the EIA, ending eleven weeks of consecutive increases.

Investors are awaiting data from the EIA due later in the day as a further gauge of the country's demand outlook.

(Additional reporting by Manash Goswami in Singapore; Editing by Keiron Henderson)