The police chief accused Mossad of being behind both threats, the state-owned paper added.
The first threat came days after Khalfan released pictures of the suspected killers and at the same time accused Mossad of involvement in the murder.
"Protect your back if you were capable of leaving your tongue loose," a message said, according to the paper.
Experts in the United Arab Emirates had traced the source of this message, Khalfan told the paper, without giving further details.
Khalfan in mid-February released the names and photographs of 11 murder suspects he said had entered Dubai with European passports -- six from Britain, three from Ireland, one from Germany and one from France.
The second threat, the paper said, was a telephone call to one of Khalfan's relatives, a retired top Emirati official, from a "Westerner with a dual passport" whom, he said, had asked "my relative to advise me to remain silent."
It was later proved that the caller was a retired Mossad agent, he added.
Khalfan, according to the report, also revealed that the authorities of a "Western country" had arrested a suspect involved in Mabhuh's assassination "two days ago."
The person arrested was among a number of suspects for which international red notices had been issued by global police agency Interpol on behalf of the UAE, he added, without giving details of the country or the person involved.
The UAE is seeking the extradition of the suspect, the daily added.
Mabhuh, a founder of the military wing of the Islamist Hamas movement which controls the Gaza Strip, was found dead in his room in the Al Bustan Rotana hotel near Dubai airport.
Twelve British, six Irish, four French, one German and three Australian passports were used by the 26 people believed linked to the murder, according to Dubai police.
In many cases, the travel documents appeared either to have been faked or obtained illegally. The countries whose passports were used all called in Israeli envoys for talks.
Britain announced in March that it was expelling one Israeli diplomat while Australia announced in May that it was throwing out an official from the Israeli embassy.
Israel has said there is no evidence linking it or Mossad to the assassination.
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