July 9 (Reuters) - The dollar gained on Tuesday after Federal Reserve Chair Jerome Powell acknowledged progress in inflation and a cooling job market, but did not give a clear signal that the U.S. central bank is close to cutting interest rates.

Powell said that inflation "remains above" the U.S. Federal Reserve's 2% target, but has been improving in recent months and "more good data would strengthen" the case for central bank interest rate cuts.

In remarks to Congress, he also noted that the job market has cooled, adding that "we now face two-sided risks," and can no longer focus solely on inflation.

But Powell stopped short of offering the dovish view of the economy that some market participants were looking for.

"The market is counting the days until we get a rate cut signal from Federal Reserve Chair Powell and I think there were some in the market who were looking for a more concrete step towards rate cuts later this year," said Adam Button, chief currency analyst at ForexLive in Toronto.

"When he didn't deliver that we saw a little bit of U.S. dollar buying," Button said.

The dollar index, which measures the U.S. currency against the euro, sterling, yen and three other major rivals, was last up 0.15% at 105.13. It fell to 104.80 on Monday, the lowest since June 13.

"I don't think he said anything that we didn't know already," said Marc Chandler, chief market strategist at Bannockburn Global Forex in New York. "The highlight would be that he says the labor market has cooled, so that's a little recognition of last Friday's jobs data."

The government's jobs report for June on Friday showed rising unemployment and downward revisions for jobs gains for the prior two months.

Traders boosted bets after the jobs data that the Fed will cut twice by December. A cut in September is seen as having a 73% probability, down from 76% on Monday, according to the CME Group's FedWatch Tool.

This week’s main U.S. economic focus will be Thursday’s consumer price index for June, which is expected to show that headline prices rose 0.1% on the month, while core prices gained 0.2%. That would put annual gains at 3.1% and 3.4%, respectively.

The euro dipped after Monday's sharp swings as investors came to terms with a hung parliament in France, which points to a potential political gridlock but lessens fiscal concerns stemming from outright far-right or leftist victories.

French political leaders from the left-wing bloc that came first in Sunday's legislative election said they intended to govern according to their tax-and-spend program, but centrists laid claim to a role as the left lacks a majority.

The single currency was last down 0.11% at $1.081. It reached $1.0845 on Monday, the highest since June 12.

The European Central Bank can continue to gradually reduce interest rates without jeopardizing a current fall in inflation, governing council member Fabio Panetta said on Tuesday.

The ECB cut rates for the first time in June from their record highs, but has made no explicit commitment on a follow-up move.

The dollar strengthened 0.29% to 161.28 Japanese yen . It is holding below a 38-year high of 161.96 reached last week.

Some market players called on the Bank of Japan to slow its bond buying to roughly half the current pace under a scheduled tapering plan due out this month, the central bank said on Tuesday.

Sterling weakened 0.15% to $1.2785.

In cryptocurrencies, bitcoin gained 2.69% to $57,776.

(Reporting by Karen Brettell; Additional reporting by Hannah Lang; Editing by Jan Harvey, Angus MacSwan and Nick Zieminski)