LONDON, June 12 (Reuters) - Euro zone bond yields fell in early trading on Friday, as they caught up with overnight remarks from President Donald Trump that a deal between Iran and the U.S. could be signed as soon as this weekend, even as Tehran said it had not made a final decision.

Germany's 10-year yield, the euro zone benchmark, dropped 4 basis points in early trading to its lowest since June 3. DE10YT=RR

Its rate-sensitive two-year yield dropped 5 bps to 2.62%. DE2YT=RR

Yields have been rising and falling in line with war headlines as traders think the longer the Strait of Hormuz stays closed and oil prices remain elevated, the greater the risk of high energy prices spilling over into broader inflation, and necessitating substantial central bank rate hikes.

Underscoring this, the European Central Bank raised interest rates on Thursday in the hope of curbing inflation before a surge in fuel costs spreads more broadly across the economy.

ECB President Christine Lagarde offered few clues as to policymakers' thinking about the coming meetings at her post decision press conference. But two sources told Reuters that policymakers see keeping interest rates on hold at their next meeting in July as the most likely scenario, if energy prices stay near their current level.

Money market bets currently reflect around a one-in-three chance the ECB hikes in July. A move in either July or September is seen as a near certainty.

Other euro zone rates were moving in line with the benchmark. Italy's 10-year yield was down 5 bps at 3.75%.