Rand Paul of Kentucky remains opposed, despite personal overtures from the president.
The full Senate is expected to vote on the nomination this week.
The committee action won't necessarily stall Pompeo's confirmation before the full Senate, but it would be an unusual setback not seen since the panel took a pass on John Bolton, President George W. Bush's pick for ambassador to the United Nations.
Trump and his allies accuse Democrats of playing politics.
Mike Pompeo's nomination for secretary of state received a boost Thursday with support from Democratic Sen.
Trump has spoken glowingly of Pompeo.
But all 10 Democrats and Republican Sen.
"When I met with Director Pompeo, we had a productive conversation about the complex security challenges we face, including the unsafe threat posed by North Korea's nuclear missile program", he said in the statement.
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As this would put the onus on businesses to collect the tariffs - in lieu of a border check collecting it - the European Union rejected it.
While he was confirmed to lead the Central Intelligence Agency with a 66-32 vote a year ago, Pompeo has faced new challenges in the Senate, with lawmakers decrying a "lack of diplomacy" they say is needed to act as the nation's top diplomat. But progressive groups are pounding senators' offices in opposition to Pompeo's hawkish foreign policy views and negative comments about gay marriage and Muslims.
As the nomination moves to the Senate floor, the GOP's slim majority, which slips with the absence of Sen. John McCain's ailing health, Pompeo may need at least some Democrats to vote to confirm him.
The Senate panel vote, however, could have been locked up for Pompeo were it not for the opposition of Paul, who said he will vote against Pompeo because he supported the 2003 American invasion of Iraq that eventually toppled Iraqi dictator Saddam Hussein. Dianne Feinstein of California, Tim Kaine of Virginia, Brian Schatz of Hawaii and Jeanne Shaheen of New Hampshire.
The two Democratic senators who have voiced support - Manchin and Heidi Heitkamp - would give Pompeo a majority of 51 votes in the 100-seat chamber, enough for confirmation.
Gardner said the president's nominees should be voted on, but he wouldn't directly answer if he supported such a move by McConnell.
Mike Pompeo could make history this week, but not necessarily in a good way.
Schatz said he is voting against confirming Pompeo because he has "an alarming tendency towards military provocation and brinkmanship".