SYDNEY, July 1 (Reuters) - The Australian and New Zealand dollars started the week on the front foot as a cooling in a key U.S. inflation reading reinforced rate cut bets, while they lost ground to a buoyant euro after France's first round election votes.

The Aussie rose 0.1% to $0.6672, having eked out 0.4% last week to as high as $0.6689 thanks to the benign U.S. PCE data, which kept the prospects of a September rate cut alive..

It still faces resistance around 67 cents, while having support at $0.6576, a range that it has held onto for the past six weeks.

The New Zealand dollar gained 0.3% to $0.6102, having bounced off the 200-day moving average of $0.6069 on Friday to break its recent declining trend. It fell 0.5% last week.

The two currencies lost ground to the euro which recovered a little from the recent selling, after the first round of the French election on Sunday showed Marine Le Pen's far-right National Rally (RN) party won the most votes but by a smaller share than some polls had initially projected.

The euro gained 0.3% to A$1.6107, moving away from a one-year trough, while it also rose 0.1% to NZ$1.7611.

The two antipodeans also hit new highs on the battered yen. The kiwi hit a 34-year high of 98.29 yen, while the Aussie scaled another 17-year top of 107.57 yen.

Down Under, Australia's home prices keep hitting record highs with another 0.7% climb last month, adding to household wealth.

The Commonwealth Bank of Australia on Monday revised up its forecast for home prices this year to an increase of 7%, from 5% previously, noting the risk still remains on the upside.

"The near term risk of an interest rate hike would limit upside risks to home prices, and even slow the pace growth," said Belinda Allen, a senior economist at the CBA.

Combined with the sticky inflation, markets are betting there is a 65% probability that the Reserve Bank of Australia will have to deliver a hike in the current 4.35% cash rate in August, which has supported the Aussie in recent weeks. .

The RBA will release the minutes of the June policy meeting on Tuesday and retail trade data are due on Wednesday.

Tax cuts, designed to give every Australian tax payer more cash to meet rising living costs, came into force on Monday.

(Reporting by Stella Qiu; Editing by Michael Perry)