* German 10-year yields fall 3 bps
* Lower-rated euro zone yields slip but lag Bunds
* Caution pending Fed meeting limits gains (Updates prices, adds more comments)
By Emelia Sithole-Matarise
LONDON, June 16 (Reuters) - Top-rated euro zone bonds pushed higher on Monday as concerns over escalating violence in Iraq drove investors to seek safe havens, though gains were limited before the U.S. Federal Reserve's policy meeting this week.
Financial markets were unsettled after Sunni insurgents seized a mainly ethnic Turkmen city in northwestern Iraq on Sunday after heavy fighting, solidifying their grip on the north after a lightning offensive that threatens to split Iraq.
German Bund futures rose 32 ticks on the day to 145.77 with yields on 10-year Bunds, the benchmark for euro zone lending, 3 basis points lower at 1.35 percent. Yields on other top-rated euro zone bonds were down by a similar amount.
"The geopolitical risk (from Iraq) is the main factor supporting core govvies (government bonds) but I don't think it will go far as we have the FOMC meeting looming which means markets will be a bit cautious," said KBC strategist Piet Lammens.
"The market is positioned for a rather dovish interest rates path going forward so there are some risks the Fed will try to make the market less complacent."
Higher oil prices due to instability in Iraq could hamper global growth and this is likely to figure in the Fed's two-day policy meeting starting on Tuesday.
It is widely expected to keep steadily reducing its massive bond-buying stimulus by $10 billion per month but investors are keen for any hints on when it might begin raising rates. Markets expect the U.S. central bank to start hiking interest rates a year from now.
Any hints the Fed might hike rates earlier than anticipated could push bond yields higher, though the impact on euro zone bonds will be tempered by the European Central Bank's recent policy easing measures.
Bunds continued to outperform their U.S. Treasuries, with the two-year U.S. T-note yield premium over German counterparts at its highest in seven years at 43 basis points, highlighting the divergence in money policy between the Fed and the ECB.
Bunds outpaced UK gilts too after Bank of England Governor Mark Carney stunned financial markets by saying rates could rise sooner than they had expected.
"We recommend investors sell five-year gilts and buy five-year Bunds, reflecting diverging policy expectations between the BoE and the ECB," RIA Capital Markets strategist Nick Stamenkovic said.
The ECB policy outlook also supported most peripheral euro zone bonds though they lagged Bunds.
Traders were also looking at developments in euro zone money markets after the overnight bank-to-bank lending Eonia rate fixed at a fresh record low on Friday, expecting it to fall into negative territory with surplus cash in the banking system set to jump this week.
The amount of cash euro zone banks have beyond what they need for their day-to-day operations and the ECB slashing its overnight deposit rate below zero a week ago are key factors holding short-term rates near zero.
The excess liquidity is predicted to rise this week above 200 billion euros ($272 billion), from just over 112 billion euros, when the ECB stops withdrawing cash to neutralise the effect of the bond purchases it made at the height of the debt crisis.
Many in the market expect Eonia to fix in negative territory this week for the first time ever - effectively meaning some banks will pay to lend other banks their cash overnight - but some said it might be short-lived because sustained negative rates would drive out money market funds.
"Low short-rates for too long could be a harmful environment for money market funds and unsecured lending volumes could suffer, increasing the fragmentation (in the euro zone banking system)," Nordea strategists said in a note. ($1 = 0.7345 Euros) (Editing by Toby Chopra/Ruth Pitchford)