* German Ifo falls more than expected in June
* Lower-rated bonds rally amid yield hunt
* Banks borrow more than expected in 1-week loans
* Money markets to be contained by new liquidity (Updates prices, adds fresh quote)
By John Geddie
LONDON, June 24 (Reuters) - Bond yields in the euro zone's lower rated states fell on Tuesday, as a second round of weaker-than-expected economic data and the promise of low-for-longer official rates spurred investor demand.
Germany's Ifo index of business sentiment eased more than expected in June to its lowest level this year, further evidence of a stuttering recovery that was highlighted by lower-than-expected private sector growth data on Monday.
The weak data underlines comments from European Central Bank President Mario Draghi over the weekend that prolonging banks' access to unlimited liquidity up to the end of 2016 was a signal on rates.
"My underlying thought is that more aggressive ECB should be more beneficial for non-core spreads," Jussi Hiljanen, chief fixed income strategist at SEB in Stockholm.
Spanish 10-year yields dropped 5 basis points to 2.65 percent, while Italy's fell 4 bps to 2.75 percent. Those in Greece plunged 17 bps to 5.82 percent, while Portugal's dived 7 bps to 3.47 percent.
Weak economic data boosted bonds this week because it was seen as increasing chances for central bank support. But it was also wearing on the credit fundamentals of member states.
S&P's head of EMEA sovereign ratings Moritz Kraemer said that much work still needed to be done in cutting debt and boosting growth in the single-currency bloc and that he saw a "calm period ahead" for ratings actions.
In the money markets, the euro overnight interbank lending rate edged up from record lows, although strategists predicted a healthy dose of new liquidity should keep rates contained.
Spot Eonia settled at 0.031 percent after markets closed on Monday, just above the record low fix of 0.01 percent last Thursday.
Banks borrowed 115 billion euros in one-week loans at the ECB's regular offering of unlimited cash on Tuesday, 17 billion more than the loans maturing this week.
This should boost the amount of cash euro zone banks have beyond what they need for their day-to-day operations to about 150 billion euros, well above the three-year low of 70 billion hit at the end of last month.
Banks tend to borrow more at these loan offerings before monthly and quarterly reporting deadlines, to show a healthy liquidity position on their balance sheets.
"Liquidity conditions should stay more or less stable ... so EONIA should stay more or less in this region although we could see the usual spike just at quarter end," Commerzbank strategist Benjamin Schroeder said.
But before the announcement, some strategists had been uncertain of the extent of the take-up given that the ECB's new negative deposit rate policy is charging banks for keeping their cash overnight.
A Reuters survey of 24 money market traders showed that banks were expected to borrow 102.5 billion euros.
The same survey estimates that banks will borrow 10 billion euros of three-month loans from the ECB at a tender on Wednesday, less than the 12.6 billion redeeming, which should serve to drain excess liquidity a fraction. (Additional reporting by Marius Zaharia; Editing by Alison Williams)