* U.S. 10-year, 30-yields on track for third weekly gain

* Focus on heavy US Treasury debt supply next week

* Fed's Williams says rate cuts likely later this year

NEW YORK, Feb 23 (Reuters) - U.S. Treasury yields tumbled on Friday from multi-month highs as investors consolidated positions after a week-long run-up, with the market having fully priced in the shift in the Federal Reserve's monetary policy outlook to a gradual easing path.

U.S. two-year yields slid from a 2-1/2-month high, while those on the three-year, five-year and seven-year notes dropped from nearly three-month peaks.

"The shift in Fed outlook to a less aggressive easing path has been priced in, and bargain hunters, short coverers are moving in after rates cheapened to multi-month highs," said Kim Rupert, managing director of global fixed income analysis at Action Economics in San Francisco.

A combination of less dovish Fed comments late on Thursday and those from European Central Bank officials earlier on Friday helped push Treasury yields higher, before they faded and traded lower.

ECB President Christine Lagarde called the relatively benign fourth-quarter wage growth data "encouraging" but not yet enough to give the central bank the confidence that inflation has been defeated.

Bundesbank President Joachim Nagel, a member of the ECB's Governing Council, backed Largarde's view. Nagel said the ECB should resist the temptation to cut interest rates early.

Their remarks pushed euro zone yields higher, and that spilled over to Treasuries.

Comments from Fed officials all week suggested the U.S. central bank will take its time in cutting interest rates to make sure that inflation falls to its 2% target on a sustainable basis.

As a result, the rate futures market has also reduced expectations of the number of rate cuts to three this year, with the start of the easing cycle possibly commencing in June or later, according to LSEG's rate probability app.

In afternoon trading, the benchmark U.S. 10-year yield was down 7.1 basis points (bps) to 4.255%. The 10-year yield, however, was on track to post its third straight weekly gain.

U.S. 30-year bond yields fell 8.6 bps to 4.377%, on pace as well to rise for a third consecutive week.

On the shorter-end of the curve, U.S. two-year yields slipped 2.2 bps to 4.691%. Earlier in the session, they hit 4.759%, the highest since Dec. 11.

On the week, the two-year yield rose 5.8 bps, on track for its fourth straight week of gains.

New York Fed President John Williams added to the chorus of central bank officials who believe rate cuts will be delayed this year despite stronger-than-expected inflation and labor market data in January.

"At some point, I think it will be appropriate to pull back on restrictive monetary policy, likely later this year," Williams said in an interview with Axios.

Investors are also focused on next week's heavy Treasury debt supply. The U.S. Treasury on Monday will sell $63 billion in two-year notes and $64 billion in five-year notes. Both are all-time highs in terms of volume. On Tuesday, the Treasury will auction $42 billion in U.S. seven-year notes.

"With all that supply, it's going to be tough for the market to go lower in yields," said Tom di Galoma, managing director and co-head of global rates trading at BTIG. "I like buying the market on back-ups in yields just because at some point the Fed will pivot and rates will be lower."

In the corporate bond space, the spreads between both investment-grade and junk-rated corporate bond yields over U.S. Treasuries have fallen to their narrowest level in more than two years, a sign of overall investor confidence is growing.

The spread on the ICE BofA U.S. Corporate Index, a commonly used benchmark for high-grade debt, declined to 93 (bps) on Thursday, its lowest since November 2021.

On the ICE BofA U.S. High Yield Index, a benchmark for junk bonds, the option adjusted spread dipped to 322 basis points, its lowest since January 2022. (Reporting by Gertrude Chavez-Dreyfuss; Editing by Paul Simao; Editing by Will Dunham)