Brent crude futures rose 6 cents to $80.17 per barrel by 0004 GMT and U.S. West Texas Intermediate crude futures were up 4 cents, or $75.79.
U.S. data on Wednesday showed consumer prices rose modestly in June, registering the smallest annual increase in more than two years. Markets expect one more interest rate rise, but oil traders hope that may be it because higher rates can slow economic growth and reduce oil demand.
Top producer Saudi Arabia pledged last week to extend a production cut of 1 million barrels per day (bpd) in August, while Russia will cut exports by 500,000 bpd.
A factor that may cap price gains was a U.S. Energy Information Administration report of a much bigger-than-expected U.S. crude stock build of nearly 6 million barrels last week.
Gasoline inventories remained largely unchanged at 219.5 million barrels during the Fourth of July holiday week, a situation that is "almost unheard of," said Phil Flynn, an analyst at Price Futures group.
Analysts had expected a big drawdown of gasoline stocks as drivers took to the roads for holiday travel.
