* Volatile trade after ECB press conference
* Draghi upbeat on economic outlook
* But adds ECB easing bias stays in place
* Euro zone bond yields, euro fall
* Euro zone banking stocks extend losses (Updates prices, adds comments )
By Dhara Ranasinghe and Patrick Graham
LONDON, April 27 (Reuters) - Euro zone government bond yields tumbled and the euro hit session lows after European Central Bank chief Mario Draghi said policymakers did not discuss removing the bank's easing bias on monetary policy at Thursday's meeting.
Bond yields and the single currency had initially jumped after Draghi said at a news conference that the recovery was "increasingly solid" and that downside risks to the euro zone's recovery had "diminished."
While that was seen by analysts as foreshadowing a bolder change at the June meeting, the ECB president also stressed the central bank had stuck to its ultra-easy policy stance as inflation continues to undershoot its target.
That was enough to persuade market participants that, after speculation this week that fading political risks in France had eased pressure on the ECB, it was in no rush to tighten policy.
"Draghi said that the balance of risks to inflation have not materially improved and this is what matters," said Chris Scicluna, head of economic research at Daiwa Capital Markets.
"Markets should not have been surprised by the comments but maybe after (Emmanuel) Macron's win and recent strong data, some in the market thought there could be a change."
Centrist Macron is expected to beat anti-euro far-right Marine Le Pen in the final round of the French presidential election next month after the two candidates emerged as the winners in Sunday's first-round vote.
Earlier on Thursday, the ECB left its key interest rates unchanged.
But in volatile trading as investors digested mixed messages from Draghi, German government bond yields gave up initial rises and fell.
They extended those falls into late trade, with the benchmark 10-year Bund yield down 5 basis points at 0.29 percent and on track for its biggest one-day fall since February.
Shorter-dated bond yields were 4 bps lower at minus 0.73 percent.
There were steep yield falls across the euro area, led by lower-rated debt in southern Europe - key beneficiaries of the ECB's bond-buying stimulus.
Portugal's 10-year government bond yield slid 12 bps to 3.47 percent, its lowest level since mid-November.
Italy's FTSE MIB and euro zone bank shares hit a day's low, last down 1.76 percent, although the broader euro zone STOXX 50 pared earlier losses.
The euro fell 0.4 percent on the day to a low of $1.0850 .
"The Q&A revealed that ... some of the members hadn't turned at all whereas I felt Draghi had shifted a little bit to be less dovish," said Neil Jones, head of FX sales at Mizuho.
"I didn't sense that was unanimous," he added, saying Draghi's comment on the easing bias had added to pressure on the euro.
Money market rates meanwhile fell as investors scaled back expectations for higher ECB interest rates in the coming months.
Forward Eonia bank-to-bank rates dated for the ECB meeting on March 8, 2018, stood at around minus 0.31 percent, about 5 basis points above the Eonia spot rate of minus 0.36 percent.
This gap suggests markets are pricing in roughly a 50 percent chance of a rate hike early next year, compared with around 60 percent earlier in the day. (Reporting by London Markets Team; Editing by Tom Heneghan)