* Graphic: World FX rates in 2019 http://tmsnrt.rs/2egbfVh
* Dollar struggles near multi-month lows vs euro, yen
* Aussie receives light lift after RBA Lowe's comments on policy
TOKYO, June 24 (Reuters) - The euro rose to a three-month high against the dollar on Monday, as bearish bets on the U.S. currency remained solid on prospects of a near-term interest rate cut by the Federal Reserve.
The euro stretched its rally last week, up 1.4%, and advanced about 0.15% to $1.1386 in early Asian trade, its highest since March 22.
The dollar index versus a basket of six major currencies was a shade lower at 96.135, having struck 96.093 on Friday, its lowest since March 21, after the Fed last week opened the door for a potential rate cut as early as next month.
That weighed on the dollar and in turn reinvigorated its counterparts like the euro, which has had troubles of its own including Italy's debt problem and the possibility of the European Central Bank having to ease policy.
"It is true that the ECB may have to ease policy especially with the Fed having shifted to an easing bias," said Yukio Ishizuki, senior currency strategist at Daiwa Securities.
"But the ECB already employs a negative interest rate policy and does not have much further room to ease even if they wanted to, unlike the Fed. It is factors like these which have seemingly supported the euro."
The dollar nudged up 0.1% to 107.400 yen after retreating to a near six-month low of 107.045 on Friday.
The U.S. currency was pressured even more against the yen, which often serves as a safe haven in times of political angst, as tensions grew between Iran and the United States.
Also in focus was whether Washington and Beijing can resolve their trade dispute at a summit in Japan this week of leaders from the Group of 20 leading world economies.
The Australian dollar was up 0.4% at $0.6952 after Reserve Bank of Australia Governor Philip Lowe said it would be legitimate to question the effectiveness of global monetary policy easing to boost economic growth.
The comments were perceived to be slightly less dovish as just last week Lowe said a recent cut in Australia interest rates to an all-time low of 1.25% would not be enough to revive economic growth. (Editing by Shri Navaratnam)