Russia wants to move talks to avoid conflict with G20 meetings in Japan.
Barrasso, from gas-producing Wyoming, seeks to hamper Russia-to-Germany pipeline.
Russia signals it may support extended supply cut with OPEC.
Russia president Vladimir Putin said last week Russia and OPEC disagreed over the fair price for oil, but that they would take a joint decision at the policy meeting.
Polish pipeline operator PERN confirmed that clean oil supplies from Russia had been partially restored on June 9 after Russia's oil export flows to Europe were disrupted in April because of contamination.
Company is betting on a sharp rise in demand for natural gas for power generation.
Speaking at an economic forum in St. Petersburg, Sechin said a third of global oil reserves were restricted by U.S. sanctions on Iran and Venezuela and that Washington was losing moral ground as a self-styled leader of open markets.
The fields are estimated to hold reserves of about 1.1 billion tonnes of oil.
Russian oil output fell to 11.11 million barrels per day (MMbbl/d) in May, its lowest level since June 2018, from 11.23 MMbbl/d in April, Energy Ministry data showed on June 2.
Deputy PM says Moscow will consider oil prices vs. loss of market share in decision.