Despite his history of business and friendship with the Russian president, Vladimir Putin, oil boss Rex Tillerson gained two key supporters among Republican skeptics on Sunday. A Democratic leader, meanwhile, hinted at battles to come over Donald Trump’s extremely conservative picks to fill the government.

Two Republican members of the Senate foreign relations committee, John McCain and Lindsey Graham, announced their support for Tillerson after weeks of expressing fears about the ExxonMobil chief executive’s history with Russia.

“Though we still have concerns about his past dealings with the Russian government and President Vladimir Putin, we believe that Mr Tillerson can be an effective advocate for US interests,” the senators said in a joint statement.

Senator Marco Rubio remains a holdout on Tillerson on the committee, which votes to approve the nomination before sending it to a larger vote on the Senate floor. Rubio has not said whether he will support Tillerson, whom he criticized sharply for failing to condemn human rights abuses in Russia, Saudi Arabia and the Philippines.

McCain and Graham abandoned Trump in the late days of the 2016 presidential campaign, and since his election they have strongly criticized his ideas about moving away from Nato allies and toward close ties with Putin.

On Sunday they again stated those views, saying: “We must strengthen our alliances and partnerships across the globe, and marshal them to defend our shared vision of world order.”

They said, however, that in private conversations and at the hearing, Tillerson had expressed views that “give us confidence that he will be a champion for a strong and engaged role for America in the world”.

In 2013, Tillerson was awarded an Order of Friendship medal by Putin for his work in Russia, and engineered a joint venture with Russian oil companies that is estimated to be worth hundreds of billions of dollars.

Tillerson is also close to Igor Sechin, according to former US ambassador to Russia Michael McFaul. Sechin is a close friend of Putin, head of one of Russia’s oil giants, and subject to western sanctions.

In his confirmation hearing, Tillerson was asked whether he would call Putin a war criminal for his order to indiscriminately bomb Chechnya in the early 2000s.

“I would not use that term,” Tillerson said. He did, however, break with Trump on a number of issues, saying that Kremlin aggression and nuclear proliferation pose real threats to the US.

He said he acknowledged that global warming has dangers but did not see it as the immediate threat that the outgoing secretary of state, John Kerry, repeatedly expressed. He refused to answer questions about ExxonMobil’s knowledge of climate change or its lobbying to remove Russian sanctions.

The Trump administration has only nominated 28 of 690 positions that require Senate confirmation, according to the Partnership for Public Service. The Senate minority leader, Chuck Schumer, hinted on Sunday of the battles to come over Trump’s more “extreme” appointments.

The Democratic leader told CNN’s State of the Union that he would for instance oppose Jeff Sessions, Trump’s pick to run the justice department. Schumer has repeatedly said he is “not confident” that the Alabama senator, based on his record, would defend American rights or act independently of the Trump White House.

Schumer said that besides the defense secretary, James Mattis, and homeland security secretary, John Kelly – retired generals respected and approved with bipartisan support – Trump’s nominees needed “thorough review”.

Schumer also said Democrats would fight to keep the ninth seat empty on the supreme court, unless Trump nominated “someone who is mainstream and could get bipartisan support”.

“If they don’t we will fight it tooth and nail, as long as we have to,” Schumer said.

Schumer’s promise echoed that of the Senate majority leader, the Republican Mitch McConnell, who last year took the unprecedented step of completely blocking Barack Obama’s nominee for the court after the death of Antonin Scalia.

Even before Obama nominated Merrick Garland, a veteran judge praised by Republicans, McConnell vowed to block any appointment through the November election. Earlier this month, McConnell claimed “the American people simply will not tolerate” any move by Democrats to block a Trump nominee.

“Apparently there’s yet a new standard now, which is to not confirm a supreme court nominee at all,” McConnell said.

On Sunday, McConnell said he was confident that any supreme court nominee would be approved by the Senate.

“I believe we’ll be able to confirm the president’s entire cabinet,” he told Fox News Sunday. “There’s great enthusiasm.”

In the same interview, McConnell criticized Democrats for insisting on more debate over Mike Pompeo, Trump’s pick to lead the CIA, “for some inexplicable reason”.

Democrats have said they want to question Pompeo about his views about collecting information about American citizens and his legal standards for using a drone to kill an American citizen.

“What’s been unfortunate is that all I asked of my colleague Senator Schumer was to treat President Trump the same we treated President Obama,” McConnell said.