WASHINGTON, July 8 (Reuters) - Concern about high inflation mounted at the U.S. central bank's meeting last month, as officials followed Federal Reserve Chairman Kevin Warsh's lead to a more stripped-down policy statement even amid concerns that price increases were broadening and might require interest rate hikes.

A few participants at the June 16-17 meeting saw a case to raise rates right away.

But the broader debate seemed evenly divided, minutes of the session showed on Wednesday, with "most participants" seeing scenarios in which inflation would fall towards the Fed's 2% target on its own, and "most" also seeing situations where it would remain high. "Almost all" of that group considered a rate increase as necessary if that occurred.

"Participants generally assessed that information received over the inter-meeting period suggested that upside risks to price stability remained elevated while downside risks to achieving maximum employment had moderated a bit," the minutes said.

Ultimately, "all participants" supported holding rates steady.

Given inflation running about double the Fed's target, "members concurred that the post-meeting statement would convey the (Federal Open Market) Committee's commitment to achieving its dual-mandate goals and emphasize that the Committee will deliver price stability," the minutes said.

Market reaction was muted, with stocks barely budging, yields on Treasuries slightly paring earlier increases, and interest-rate futures maintaining bets the Fed will deliver a rate hike by its September meeting.

The policy debate and decision formed only one aspect of the discussion at Warsh's inaugural meeting as Fed chief. 

Policymakers also considered his proposal to end "forward guidance" and provide less commentary in the statement on coming rate decisions.

"A majority of participants remarked that they saw advantages in shortening the statement," the minutes said, while "most participants" supported removing language indicating the Fed's next policy move would likely be a rate cut.

The alternative approved by the Fed in June removed guidance about rates altogether, in alignment with Warsh's overall desire to avoid making promises about rate decisions.

The minutes also noted that Warsh described his plan to establish five task forces to examine how monetary policy is conducted, but do not indicate any discussion of it.

The Fed left its benchmark interest rate unchanged in the 3.50%-3.75% range at the June meeting, though new projections revealed a growing sentiment around a likely rate hike this year, with nine of 18 policymakers seeing rates slightly higher by the end of 2026.

The meeting was attended by Warsh's two new special advisers, Paul Winfree and Daniel Heil.