WASHINGTON, Jan 3 (Reuters) -

U.S. Federal Reserve officials appeared increasingly convinced at their meeting last month that inflation was coming under control, with "upside risks" diminished and growing concern about the damage that "overly restrictive" monetary policy might do to the economy, according to minutes of the central bank's Dec. 12-13 policy meeting.

As a result "almost all participants indicated that...a lower target range for the federal funds rate would be appropriate by the end of 2024," the minutes said, with "a number of participants" highlighting increased uncertainty about how long strict monetary policy would need to be maintained given the progress achieved on lowering inflation.

"A few" Fed officials said they felt the Fed was approaching a point where the central bank may face a "tradeoff" between its dual goals of controlling inflation and maintaining high rates of employment - the sort of sacrifice policymakers have hoped to avoid in their search for a "soft landing" from the worst breakout of inflation in 40 years.

"Participants pointed to the decline in inflation seen during 2023, noting the recent shift down in six-month inflation readings in particular," the minutes said, which through November were running just below the Fed's 2% target.

The minutes shed little light on when rate cuts might commence.

Indeed participants also noted "an unusually elevated degree of uncertainty" about the outlook, with further interest rate increases still possible.

But the detailed account of the session also reflected a debate underway about how to safeguard the economy while also seeing inflation continue to fall.

"Most" felt that monetary policy was having its intended impact and would continue to do so by dampening household and business spending and pulling inflation back to target.

Participants "stressed...that it would be appropriate for policy to remain at a restrictive stance for some time until inflation was clearly moving down sustainably toward the Committee's objective."

But they also said coming decisions would be "careful and data-dependent," cognizant of developing risks to the economy and faster-than-anticipated gains on inflation.

The Fed held its benchmark overnight interest rate steady in the current 5.25%-5.50% range at last month's policy meeting, with new economic projections showing most officials expect that the policy rate will need to be lowered by three-quarters of a percentage point over the course of 2024.

Precisely when the Federal Reserve will start cutting interest rates stands as the big unknown for markets and economists at the start of 2024. Market pricing shows investors currently betting the Fed will cut rates beginning at its meeting in March while economists on balance see the Fed holding off until closer to mid-year.

The Fed next meets on Jan. 30-31.

(Reporting by Howard Schneider; Editing by Dan Burns and Andrea Ricci)