The statement did not name the source but said it was planning legal action.
However, a source at the broadcaster said the threats emanated from Syria, which has been rocked by protests calling for the ouster of President Bashar al-Assad since mid-March.
"I began receiving emails swamped with curses and threats nearly three weeks ago," Tunisian presenter Laila al-Shayeb told AFP.
Shayeb, who presents Al-Jazeera's daily evening programme "Hassad al-Yawm" (the day's harvest), said the last email she received on Saturday carried a direct death threat.
"Look into the camera in front of you and behind it you will find a machete preparing to harvest your brain and decorate the walls with it," she said.
At the end of the email, the sender posted an "indecent" fabricated photo of Shayeb threatening to post a full similar video on the Internet, she said.
Another Al-Jazeera journalist, Gahda Oweis from Lebanon, told AFP she had received similar messages.
Meanwhile, a presenter who spoke on condition of anonymity said: "I was forced to cancel my annual vacation which I usually spend in my country" due to the daily threats he had been receiving.
Al-Jazeera has held meetings with its journalists to reassure them of their safety and that of their families, said the source at the news channel.
The broadcaster said in its statement that "these continuous pressures against it and against its journalists will only make it more determined to hold on to the professional editorial line it has followed.
Syrian authorities accuse Al-Jazeera and other international satellite channels of exaggerating the protests and of broadcasting footage without verifying their authenticity.
But Shayeb rejected such accusations, saying: "We deal carefully with the material we receive. We only air videos which we have managed to verify."
Because few foreign journalists gain entry to Syria, international media rely heavily on video footage filmed and released by the protesters themselves on Internet sites such as YouTube.
Rights groups say that security forces have killed more than 1,300 civilians and arrested at least 12,000 since the anti-government protests began.
The pan-Arab satellite television channel has been in hot water with several autocratic Arab regimes over its coverage of uprisings sweeping the region since January.
During the protests in Egypt that toppled president Hosni Mubarak, the channel was banned from operating inside the country and nine of its journalists were briefly detained.
In Libya, Al-Jazeera cameraman Ali Hassan al-Jaber was killed on March 12 in an ambush near Benghazi that the rebels blamed on Moamer Kadhafi's forces, and several Al-Jazeera journalists have also been arrested covering the revolt.
Also in March, Yemen evicted Al-Jazeera after it said footage of torture in an Iraqi jail was broadcast as having been filmed in the Arabian peninsula state, also the scene of anti-regime protests.
Since its launch 14 years ago, pan-Arab satellite channel Al-Jazeera has become a major player in the Arab world, giving a voice to opposition figures of various backgrounds.
Founded in gas-rich Qatar in 1996, Al-Jazeera -- with apparently limitless resources -- has revolutionised the Arabic-language media and reporting on the Middle East, a region where state channels have dominated.
Last May, Bahrain also banned the news channel.