The new, unpaid post comes as the Liberal Democrats step up their campaign for the UK to stay in the EU ahead of a pledged referendum in 2017.
Mr Moore will have support from civil servants in the Deputy Prime Minister's Office and be expected to prepare a report by July.
The Border MP was replaced as Scottish Secretary by Alistair Carmichael.
Mr Clegg described it as a "very important role".
Mr Moore added he was "pleased to play my part to ensure the government is doing all it can to support business efforts to build a secure economic future for the UK in the EU".