By Leika Kihara

TOKYO (Reuters) -At least two of the Bank of Japan's nine board members called for an early interest rate increase at a policy meeting in June, minutes showed on Monday, underlining the central bank's hawkish tilt that provides scope for further hikes ahead.

"Members agreed that the yen's recent falls were among factors that push up inflation, and must warrant close attention in guiding monetary policy," the minutes showed.

The discussions underscore how yen moves and concerns over an inflation overshoot were key factors discussed at the BOJ's June meeting, and led to its decision in July to raise interest rates to levels unseen in 15 years.

With the Japanese currency having sharply reversed course to hit a 7-month high on Monday, markets are focusing on BOJ Deputy Governor Shinichi Uchida's speech on Wednesday for clues on the pace of future rate hikes.

The yen's latest surge has been buoyed by weak U.S. labour data, which has stoked recession worries.

"Given how latest yen rises are reducing the risk of an inflation overshoot, we expect the BOJ to hike rates at a cycle of once every six months," rather than at a more frequent pace, said Yoshimasa Maruyama, chief market economist at SMBC Nikko Securities.

While the BOJ kept interest rates steady at the June meeting, some board members warned that rising import costs from a weak yen were hurting consumer sentiment and heightening the risk of an inflation overshoot, the minutes showed.

One member said the BOJ must consider "adjusting the degree of monetary easing" to forestall future risks of an inflation overshoot, given how firms are renewing efforts to pass on increasing costs to consumers.

"Another member said the BOJ must continue to closely monitor relevant data in preparation for the next meeting in July and, if deemed appropriate, raise interest rates without delay," according to the minutes.

The yen, which hovered around 157 to the dollar at the time of the June meeting, hit a 38-year low below 161 in July - a move that likely affected the BOJ's decision to hike short-term rates to 0.25% from 0-0.1% at the July 30-31 meeting.

The Japanese currency stood at 145.15 on Monday.

Following the rate hike in July, BOJ Governor Kazuo Ueda did not rule out another increase this year and stressed its readiness to keep hiking borrowing costs to levels deemed neutral to the economy - seen by analysts as anywhere between 1% and 1.5%.

The outlook for private consumption likely holds the key to the next rate hike timing, some analysts say, with rising living costs hurting household sentiment.

"Some members said close attention must be paid to how much boost consumption would get from improvements in income" from wage increases, the June minutes showed.

Japan's second-quarter gross domestic product (GDP) data, due on Aug. 15, will likely show consumption rebounding after four quarters of declines, according to a Reuters poll.