A widely anticipated inflation report on Thursday may solidify expectations for the Federal Reserve to cut interest rates in coming months.

The consumer price index, or CPI, report for June is due out at 8:30 a.m. ET. Recent economic releases have suggested that inflation and economic growth are both cooling, including last week's report that unemployment in June ticked up to 4.1%.

Thursday's report comes after Federal Reserve Chair Jerome Powell delivered two days of testimony on Capitol Hill this week. The central bank chief did not indicate when exactly rate cuts will begin. However, Powell did say the Fed sees the risks to the economy as more in balance between inflation and recession and that the central did not need to wait until inflation hit the 2% level to cut rates.

Economists surveyed by Dow Jones are looking for CPI to rise 0.1% month over month, and 3.1% year over year. The core CPI, which strips out more volatile food and energy prices, is expected to rise 0.2% from May and 3.4% since June last year.

In May, CPI was unchanged month over month and up 3.3% on an annual basis.

Focusing on the trends of unemployment and inflation could bolster the case for rate cuts, said Matt Brenner, managing vice president, investments and product management at MissionSquare Retirement.

"The level on inflation is still elevated relative to the Fed's [2%] target. The level on unemployment is still very low historically at 4.1%. But the trend in both is that unemployment is gradually starting to pick up and that inflation continues its downward trajectory," said Brenner.

"For some time the Fed has been more focused on levels, and now it seems that they may be starting to tilt more towards a focus on trend. And if that's the case, then the chances of a rate cut go up," Brenner added.

The price changes in the components that make up the CPI index will also be a focus on Thursday, especially if the number comes in different from expectations. Shelter and medical care services could be key areas to watch, said Wilmington Trust Chief Investment Officer Tony Roth.

Both shelter and medical services are also key parts of the personal consumption expenditures index, the Fed's preferred inflation measure, rather than CPI.

"We've seen medical services [be] pretty tame, and that's important because medical services makes up a much bigger portion of the PCE, which is the more important of the two inflation prints," Roth said.

The CPI report comes as markets are on the upswing.

Stocks and bonds have both rallied in July as traders grow more confident in a rate cut sometime this year. The S&P 500 crossed 5,600 for the first time on Wednesday.

Fed funds futures pricing shows traders are expecting the Fed to hold rates steady at its meeting later this month, and then cut in September, according to the CME FedWatch Tool. A month ago, the chances of another pause in September were close to a toss-up, according to the same tool, which uses 30-day fed funds futures to come up with implied probabilities.

The expected hold in July could keep Thursday's CPI report from being a big market mover, Bank of America rates strategist Meghan Swiber said in a note to clients Wednesday.

"Cooling activity and limitations on near-term cut pricing should confine market response in either direction," Swiber said.

However, Wilmington Trust's Roth said stocks could rally if the inflation reading is cooler than expected because some investors have not shaken their fears from earlier this year, when inflation briefly ran hotter.

"I don't think that the market has fully appreciated the weakness in the economy, or the fact that inflation is clearly in the rear view mirror," Roth said.

— CNBC's Michael Bloom contributed reporting.