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 areas will get access to £95 million for regenerating historic and heritage sites as part of the Future High Street Fund