The President-elect has named Rex Tillerson as his choice of top diplomat but his close links to Russian President Vladimir Putin have raised concerns.
Mr Tillerson, the outgoing ExxonMobile chief executive, has had extensive business dealing with Russia and was awarded Russia’s Order of Friendship by President Putin in 2013.
The US Senate has to vote to approve Mr Tillerson’s appointment and although many Democratic senators have expressed doubt, Senator McCain has now indicated that some Republicans are worried too.
He said: “I and several of my colleagues have concerns about Mr Tillerson, and some of his past activities, specifically his relationship with Vladimir Putin.
“I have concerns but at the same time I’m certain we will give Mr Tillerson an opportunity to make his case about why he is qualified to be Secretary of State.”
Senator McCain made his comment during a visit to the Baltic states with fellow US Senators Amy Klobuchar and Lindsey Graham.
The trio also demanded answers to claims from the CIA and FBI that Russian hackers tried to swing the US presidential election in Mr Trump’s favour.
Senator Graham, a Republican, said: “You can expect that the US Congress will investigate the Russian involvement in our elections and I predict there will be bi-partisan sanctions coming that will hit Russia hard, particularly Putin as an individual.
“It is now time for Russia to understand – enough is enough.”
Senator McCain described President Putin as a “thug and a murderer”.
Russia has dismissed the threat of sanctions as a “provocation directed by the White House”.
Foreign Ministry spokeswoman Maria Zakharova said: “To be honest, we are tired of lies about the ‘Russian hackers’, which is being poured down in the United States from the very top.”
President-elect Trump said: “I think we ought to get on with our lives. I think that computers have complicated lives greatly. The whole age of computer has made it that nobody knows exactly what’s going on.”