Christopher Steele, who runs a London-based intelligence firm, is believed to have left his home this week.
The memos contain unsubstantiated claims that Russian security officials have compromising material on Mr Trump.
The US president-elect said the claims were "fake news" and "phoney stuff".
Mr Steele has been widely named as the author of a series of memos - which have been published as a dossier in some US media - containing extensive allegations about Mr Trump's personal life and his campaign's relationship with the Russian state.
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