SYDNEY, Sept 10 (Reuters) - Three-year Australian bond yields fell to a four-month low on Thursday while New Zealand's was near zero amid expectations central banks in both countries will have to ease policy further to get their flagging economies back on track.

Yields on three-year Australian government bonds went as low as 0.232% earlier in the session, a level last seen in mid-May. They were last at 0.265%.

In New Zealand, yields on two-year bonds were at 0.03% after sliding into negative territory on Wednesday for the first time.

Indicating how long investors expect rates to remain low in New Zealand, the five-year rate was also barely positive at 0.05%.

Analysts said this was a sign markets were pricing in the probability for the official cash rate to turn negative in New Zealand.

Despite prospects for negative rates, the country's government bonds were in solid demand at an auction on Thursday with the NZ$50 million offer receiving bids worth NZ$90 million.

The two-year April 2022 bonds were successful at 0.16%-0.18% while the 2029 bonds received bids around 0.81%-0.83%.

"Overnight interest rate swaps pricing continues to tread deeper into negative territory, and there is plenty of scope for swap and bond rates to grind lower from here, taking more of the curve sub-zero," analysts at the Bank of New Zealand wrote in a note.

Data due next week will likely show New Zealand's economy shrunk more than 20% in the June quarter with a fresh coronavirus lockdown in the most populous city of Auckland casting a shadow over V-shaped recovery hopes.

In Australia, some analysts are predicting another contraction in its economy in the current quarter with the second-most populous city of Melbourne in a hard lockdown to fight a second wave of infections.

Australia's A$2 trillion economy slipped into recession in the June quarter for the first time in three decades though the blow was largely cushioned by government and central bank largesse.

In currencies, the Australian dollar came off a two-week trough to be last at $0.7280.

The New Zealand dollar was at $0.6688 from Wednesday's two-week low of $0.6602. (Editing by Kim Coghill)