Turkish media said the arrests include senior bureaucrats and police officers.
Mr Gulen was once an ally of President Recep Tayyip Erdogan, who was resoundingly re-elected on Sunday. However, the two fell out, and Mr Gulen now lives in self-imposed exile in the US.
A Turkish court issued an arrest warrant for the influential cleric in 2014, accusing him of establishing and running an "armed terrorist group" - a claim the cleric strongly denies.
The state-run Anadolu Agency said those detained in Tuesday`s raids were suspected of acting "beyond their legal authority".
Mr Erdogan has decried the work of Mr Gulen`s followers as that of "a state within a state".
He has been seeking to curb the reach the Hizmet ("Service") movement since 2013.
Last week Turkish police stormed the headquarters of a media group linked to Mr Gulen, raising concerns about media freedom days before the country`s election.
The election on Sunday followed an earlier, inconclusive poll in June, in which Mr Erdogan`s AKP party lost its majority in parliament.
Increasing tensions
Mr Erdogan`s re-election on Sunday comes amid concerns about rising tensions in the country.
Turkey has stepped up operations against the so-called Islamic State group since it was blamed for twin blasts that killed more than 100 people in Ankara last month.
The double bombing, targeting a Kurdish peace rally, was the country`s most deadly, and came amid an upsurge of fighting between Turkish forces and Kurdistan Workers` Party (PKK) rebels.
On Monday the Turkish military said it had carried out air strikes on PKK militant bases across the border in northern Iraq - the first on PKK targets in Iraq since Sunday`s election.
In a statement on Tuesday the military said that Turkish jets hit PKK shelters, bunkers and weapon storages in six locations, including the Qandil mountains where the rebel leaders are based.
The military has been targeting bases and hideouts of the PKK since July, when renewed fighting between the rebels and Turkish security forces broke out, signalling a breakdown in a two-year ceasefire.
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