* Loonie touches six-day low at 1.2127 * Price of U.S. oil increases 0.5% * Canadian bond yields rise across steeper curve TORONTO, June 10 (Reuters) - The Canadian dollar was little changed against its U.S. counterpart on Thursday as oil prices rose and investors weighed data showing rising U.S. consumer prices, with the currency holding near an earlier six-day low. The loonie was trading nearly unchanged at 1.2111 to the greenback, or 82.57 U.S. cents, having touched its weakest intraday level since last Friday at 1.2127. The price of oil, one of Canada"s major exports, rose after slipping the previous day on data indicating weak U.S. driving season fuel demand. U.S. crude prices increased 0.5% to $70.31 a barrel. The number of Americans filing new claims for unemployment benefits fell last week to the lowest level in nearly 15 months, while consumer prices increased further in May as the pandemic"s easing grip on the economy continues to boost domestic demand. Bank of Canada Deputy Governor Tim Lane is due to speak to Western Canadian financial advisers at 1 p.m. EDT (1700 GMT). On Wednesday, the central bank left its key interest rate unchanged at a record low 0.25%, as expected, and said the economy would "rebound strongly" as COVID-19 vaccinations picked up. Canadian government bond yields were higher across a steeper curve, tracking the move in U.S. Treasuries. The 10-year yield rose 3.1 basis points to 1.442%, having rebounded from its lowest intraday level in nearly three months on Wednesday at 1.403%. (Reporting by Fergal Smith Editing by Paul Simao) Our Standards: The Thomson Reuters Trust Principles.