Mr Moore is said to be the only Lib Dem Cabinet-level casualty planned, with Deputy Prime Minister Nick Clegg thought to want a more attacking approach in the run-up to the referendum.
The Prime Minister announced the appointment of Mr Carmichael on Twitter, also confirming that he will be making wider changes later.
He is expected to promote women and northern MPs in what has been branded a "flat cap" reshuffle.
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The Penrith and The Border MP received 37 votes.
In the first of the series, meet Walaa who left Syria with her family when war broke out.
An estimated 5.7 million people have fled the Syria, according to the United Nations.