(Adds details, background)

* May core orders -3.2% m/m vs forecast 0.8%

* Core orders seen as gauge of strength in private investment

* Private demand hold the key to positive growth and BOJ exit

TOKYO, July 11 (Reuters) - Japan's core machinery orders unexpectedly fell for a second straight month in May, government data showed on Thursday, highlighting the fragility of the economy.

The volatile yet leading indicator of capital spending fell 3.2% month-on-month in May, following a 2.9% drop in April and confounding a 0.8% increase seen by analysts in a Reuters poll.

The slowdown in machinery orders may be a setback for the Bank of Japan's plans to normalise monetary policy as the BOJ has embarked on unwinding its unconventional policy. It raised rates in March for the first time since 2007 and decided in June to cut government debt buying.

The Cabinet Office, which compiles the data, cut its view on machinery orders, saying there are signs that a pick-up is stalling.

The core orders exclude ship buildings and repairs as well as electricity power generation, both of which tend to volatile. Orders from overseas are also not counted as such but they are categorised as external orders, or exports. External orders make up for around 40% of overall orders, while domestic core orders account for 30%.

Compared with a year earlier, core orders, regarded as an indicator of capital spending in the coming six to nine months, increased 10.8% in May.

A Cabinet Office survey showed core orders grew 4.4% in January-March from the previous quarter, but they were expected to fall 1.6% in the second quarter.

Capital spending is one of a few bright spots in Japan due to demand for labour-saving technology, digital and green transformation to enhance labour productivity and cope with chronic labour shortages.

The government is aiming for nominal domestic investment, including research and development, to top 100 trillion yen ($619.08 billion) by the fiscal year 2028.

Gross domestic product (GDP) data showed earlier this month that private non-residential investment fell 0.4% quarter-on-quarter, making capital spending and consumer spending the major culprits behind a sharper than expected first-quarter economic contraction. ($1 = 161.5300 yen) (Reporting by Tetsushi Kajimoto; Editing by Kim Coghill)