These attempts are gaining intensity and taking various forms.
A later tweet said Doha-based media platforms were undergoing "continual hacking attempts".
It comes amid a diplomatic crisis with Qatar, which other Arab states accuse of having ties with terrorism.
The Qatari-owned news organization claims that its website and broader media network are now "undergoing systematic and continual hacking attempts".
"There were attempts made on the cyber security of Al Jazeera but we are combating them and now all our entities are operational", said a senior employee who declined to be named.
Mostefa Souag, the network's acting director-general, told Reuters that Al Jazeera will maintain editorial independence despite the diplomatic rift.
A Federal Bureau of Investigations (FBI) team has been in Doha for the past week after the Qatari government asked the United States for help following the security breach by hackers.
Last month hackers gained access to the website of Qatar News Agency, the official media outlet of the Middle East nation, and reportedly planted false statements attributed to Qatari emir Sheikh Tamim bin Hamad al-Thani.
Subsequently there were media reports suggesting that Qatar had been targeted by Russian hackers - a claim dismissed by Moscow. "We don't interfere in anybody's business, we just report", he said. "No. Al Jazeera's editorial policy is going to continue the same regardless of what happens with this event".