WASHINGTON — House Speaker John Boehner on Friday wouldn’t rule out higher income-tax rates as part of an agreement to avert the coming “fiscal cliff,” but he also said no such movement was possible unless President Barack Obama showed more interest in compromising.

The Ohio Republican was asked at a news conference whether he could accept higher rates, perhaps 37 percent, and “protect small business at the same time.”

“There are a lot of things that are possible to put the revenue that the president seeks on the table,” he said.

Boehner reiterated later that he still opposes higher tax rates, but his initial comments are significant, because they suggest that higher rates might be part of a deal.

The comments also reflect the murky nature of such negotiations, and how sending signals can be oblique, but significant.

Bush-era tax cuts expire at the end of the year, and $109 billion in automatic spending cuts will take effect Jan. 2 unless lawmakers act.

Obama campaigned on a pledge to raise taxes on the wealthy but to keep Bush-era rates intact for individuals who make less than $200,000 a year and families earning less than $250,000.

He has said over and over since the election that other rates must go up. Currently, the top rates are 33 and 35 percent; they are slated to rise to 36 and 39.6 percent next year.

Obama has suggested that he could accept a number in between, and Capitol talk has centered on 37 percent.

Boehner was hardly optimistic Friday. “None of it is going to be possible if the president insists on his position, insists on ‘my way or the highway,’ “ he said.

Republicans in the House of Representatives offered a deficit-reduction package Monday, but haven’t received a White House counteroffer. It included $800 billion in new revenue, with no specifics.

Some veteran Republican lawmakers suggested that they could accept higher rates, notably Rep. Tom Cole of Oklahoma, who is considered close to Boehner, and Sen. Tom Coburn of Oklahoma, who is regarded as one of the Senate’s most outspoken conservatives.

Obama and Boehner spoke this week, and Boehner called the conversation “pleasant, but just more of the same.”

So, Boehner said Friday, “There’s no progress to report,” adding, “The White House has wasted another week.”

White House officials declined to comment, while the administration continued its public push for middle-class tax breaks.