(Adds comments on interest rate outlook, background)

LONDON, March 23 (Reuters) – The dollar index is not far from fair value but it is unclear how much more the U.S. currency will strengthen against the euro, St Louis Federal Reserve President James Bullard said on Monday.

Speaking on CNBC, Bullard pointed to the run up to the European Central Bank’s launch if quantitative easing earlier this month as a driver of the dollar’s 25 percent surge against the euro.

“It (QE) is a major development on global financial markets and (that) the run up to that led to a depreciating euro and a strengthening dollar is not surprising at all,” he said.

“That has all been priced into markets so I think it’s unclear at this point where the dollar euro exchange rates are going to go.”

He also said an exit of Greece from the euro would be possible but prove very painful for Greece itself.

“I think it’s (Grexit) much more manageable than it would have been a couple of years ago and so I do think it could be done but I don’t think it’s advisable to try to go down that path,” he told CNBC in an interview.

“Greece would face a very different and I think bleak future if it went in that direction but I don’t think the spillover into international markets is as it would have been.”

The Federal Reserve last week moved a step closer to hiking rates for the first time since 2006, but downgraded its economic growth and inflation projections, signaling it is in no rush to push borrowing costs to more normal levels.

Bullard, who has long called for the Fed to raise rates sooner rather than later, said he had not changed his view in general, just altered the path of his own rate projection to reflect that policy had not moved as fast as he had expected.

“I’ve not become more dovish it was just the committee didn’t move in March,” he said. “I think there was some dovishness (from the meeting) because there’s doubts. Going into the summer, we are going to see how the data comes in and we’ll be able to make a move but we don’t have to make a move.” (Reporting by Anirban Nag, Karin Strohecker, Marc Jones, Emelia Sithole-Matarise; writing by Patrick Graham Editing by Jeremy Gaunt)