(John Kemp is a Reuters market analyst. The views expressed are his own)

LONDON- Hedge funds have started to temper their bullishness towards oil after crude futures prices have doubled since late April.

Crude prices are nearing levels expected to see some shale production restart and there are concerns the rally is outrunning the recovery in demand.

Hedge funds and other money managers purchased the equivalent of just 6 million barrels in the six major petroleum futures and options contracts in the week to June 2.

Portfolio managers have bought petroleum derivatives in nine out of the last ten weeks, increasing their bullish position by a total of 324 million barrels since late March

But purchases last week were the smallest for nine weeks (if the single week when funds were actually net sellers is excluded), indicating the buying wave may be fading.

Position changes were mostly insignificant with purchases of NYMEX and ICE WTI (+7 million barrels) and European gasoil (+8 million) but sales of Brent (-2 million) and U.S. gasoline (-8 million) and no change in U.S. diesel.

But there were some tentative indicators the rally in crude prices may be running out of steam, while funds try to anticipate an improvement in the relative price of distillates, where margins have become uneconomic:

Portfolio managers added new short positions in petroleum last week and their ratio of bullish longs to bearish shorts dipped for the first time since the end of March.

Fund buying in WTI was some of the smallest since portfolio managers started to accumulate positions in U.S. crude in March. Funds were net sellers of Brent for only the second time in nine weeks.

Funds have been net buyers of European gasoil for three weeks running, and last week's purchases were the largest yet, indicating traders are trying to position themselves ahead of an anticipated improvement in distillate margins.

None of these is a strong signal and there is a big risk of over-interpreting them. But taken together they suggest a possible pause or even a future reversal in fund buying.

Benchmark Brent futures prices have already more than doubled over the last seven weeks, to more than $40 per barrel from just over $20 in late April, increasing the probability of short-term profit-taking and a pull back.

At the same time, the refining margin for making gasoil from Brent has halved from $10 to $5 per barrel, and is down from $17 at the start of the year, which suggests distillates will eventually outperform.

(Editing by David Evans) ((john.kemp@thomsonreuters.com and on twitter @JKempEnergy))