HOUSTON — Exxon Mobil is pursuing a waiver from Treasury Department sanctions on Russia to drill in the Black Sea in a venture with Rosneft, the Russian state oil company, a former State Department official said on Wednesday. An oil industry official confirmed the account.

The waiver application was made under the Obama administration, the officials said, speaking on condition of anonymity, and the company has not dropped the proposal.

The proposal is now before the Trump administration at a delicate time in Russian-American relations, with rising tensions over the war in Syria and a looming congressional inquiry into reports of Russian efforts to influence the United States presidential election.

The appeal did not come up during Senate confirmation hearings for Secretary of State Rex W. Tillerson, who was Exxon Mobil’s chief executive before his nomination by President Trump and was known to have a strong working relationship with President Vladimir V. Putin of Russia. At the time, Mr. Tillerson and other company officials said they had not lobbied against the sanctions, which were imposed on Russia in response to its military intervention in Ukraine.

But Mr. Tillerson and company officials did note that Exxon Mobil had received a waiver to complete drilling of an exploration well in Russia’s Arctic waters. Company officials also disclosed that they had urged Obama administration officials to make American sanctions consistent with European Union sanctions, which gave greater latitude to European companies to continue taking part in some Russian projects.

The Exxon Mobil waiver request for the Black Sea was reported Wednesday by The Wall Street Journal.

Asked about the waiver application, Alan Jeffers, an Exxon Mobil spokesman, said, “We don’t comment on ongoing issues.” A Treasury representative said the department would not comment on individual licenses or waiver requests.

The oil industry official said that the application had been made in 2015, with Exxon Mobil arguing that it could lose its contractual exploration rights in the Black Sea if it did not begin drilling operations by the end of 2017. European companies, particularly Eni, the Italian oil and gas giant, could then pick up the work.

The Russian news media quoted Zeljko Runje, Rosneft’s vice president for offshore projects, saying last June: “We are going to drill with Eni next year. That is our plan. On the Black Sea.”

European sanctions exempted some contracts previously signed by European companies, while United States sanctions drew a harder line, international energy specialists said.

Hal Eren, a former official in the Treasury Department’s Office of Foreign Assets Control, said that such waivers were rarely requested or granted and that in most cases such permission was given only for environmental or safety reasons. The Exxon request is particularly unlikely to succeed, he said, because of the narrow nature of the current sanctions.

“I don’t think they would issue a license, especially given the political context in which this takes place,” Mr. Eren said.

United States and European sanctions were imposed on Russia in March 2014 in response to Moscow’s annexation of Crimea from Ukraine. Even as the Ukraine crisis deepened, Exxon continued pressing for deeper involvement in Russia’s oil industry. 

The company’s exploration chief, Neil W. Duffin, signed an expansion of its joint venture projects in the spring, even after Rosneft’s chief executive, Igor I. Sechin, had been personally blacklisted under the sanctions. The deal violated no laws, but was made despite the clear risk of wider sanctions over the war in Ukraine.

Tighter sanctions on advanced oil drilling technology transfers and on Rosneft itself were enacted a few months later when Russia was implicated in the shooting down of Malaysia Airlines Flight 17 over eastern Ukraine that July.

Exxon Mobil has received a few exceptions to the sanctions.

It was allowed to complete drilling from one advanced rig in the Kara Sea, in the Arctic, in September and October 2014, after arguing it would not be safe for the environment or the equipment to leave the well unfinished. No oil was produced, but oil was found for future production.

As Exxon Mobil’s chief executive, Mr. Tillerson took issue with the initial sanctions, before the tightening in late 2014. At Exxon’s 2014 annual meeting, he said, “We do not support sanctions, generally, because we don’t find them to be effective unless they are very well implemented comprehensively, and that’s a very hard thing to do.”

As secretary of state, Mr. Tillerson has criticized Russian policies, most recently Mr. Putin’s continuing support of the Syrian government. He has not suggested that any sanctions should be lifted.

Mr. Tillerson has pledged to stay clear of decisions that will affect his former employer.

Exxon’s waiver request drew criticism Wednesday on several fronts. Senator John McCain, the Arizona Republican who is a fierce critic of Mr. Putin, asked in a Twitter post, “Are they crazy?”

Michael A. McFaul, who was United States ambassador to Russia under Mr. Obama, said on Twitter that if the Trump administration granted a waiver, “then all that tough talk last week about Russia was just that — talk.”

Exxon Mobil was particularly active in Russia during the Obama administration’s early attempts to “reset” relations, which helped reopen Russian oil fields for foreign investment. Exxon signed a deal to explore in the Black Sea in early 2011, with the promise to share in the prize if any oil were found.

Black Sea offshore drilling rights also were included in a sweeping strategic partnership that Exxon, under Mr. Tillerson, signed with the Russian state oil company later that year.

The Black Sea is a deepwater prospect potentially rich in reserves, though so far exploration has been fairly limited. The proposed drilling block is just east of Crimea but not in waters disputed by Ukraine.

It is only one of several areas in Russia that Exxon Mobil hopes to develop to expand its reserves for future business. Its interests in the Barents Sea and the Bazhenov shale field are essentially closed to development because of the sanctions.

The company continues to drill in more conventional fields around Sakhalin Island and around Siberia, which are not covered by sanctions against technology transfers for unconventional drilling in Arctic seas and shale fields.

In their August 2011 strategic partnership deal, Exxon and Rosneft agreed to invest $3.2 billon in exploration in the Arctic Ocean and the Black Sea, in an area known as the Tuapse Trough license block. The area covers 2.8 million acres and plunges to depths of 6,000 feet, according to Rosneft’s website.