The arrested man was a constable from Vehari city in Punjab, officials were quoted as saying by TV news channels.
The officials said the man was arrested after they traced a phone number that was used to make threatening calls and send SMSes to Interior Ministry officials after terrorist attacks in Rawalpindi and Karachi on November 21.
Interior Minister Rehman Malik had told the media last week that he and some officials had received threats on their mobile phones.
He had further said that the sender of the threats had been traced.
Media reports said the arrested constable was linked to the banned Sipah-e-Sahaba, a notorious sectarian group, and that he was allegedly involved in terror attacks.
There was no official word in this regard.
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Pakistani authorities have put in place extensive security arrangements to prevent terror attacks on Shia processions and prayer halls during the holy month of Muharram.
Bans have been imposed on riding motorcycles or riding pillion and cellular services have been suspended in nearly 50 towns and cities across the country.
Despite the security arrangements, nearly 40 people have died in terror attacks targeting Shias since last week, including 23 killed in a suicide attack on a procession in the garrison city of Rawalpindi on Wednesday.