Twenty unidentified vessels were seen approaching the flotilla earlier on Wednesday night, multiple people on board said, as passengers put on life vests and braced for a takeover.
"Multiple vessels ... were illegally intercepted and boarded by Israeli Occupation Forces in international waters," the organisers said in a statement.
"We are diligently working to account for all participants and crew."
It said that its communications were jammed before boarding began, which interfered with cameras that were providing live streams from various boats and communications between vessels.
Several vessels were stopped and their passengers are being transferred to an Israeli port, the Israeli foreign ministry said in a statement on Wednesday.
Greta Thunberg, the Swedish climate campaigner, and her friends are "safe and healthy," the foreign ministry said in a post on X alongside a video that appeared to show Thunberg and several masked and armed Israeli military personnel.
The Global Sumud Flotilla, which consists of more than 40 civilian boats carrying about 500 parliamentarians, lawyers and activists including Thunberg, is trying to break Israel's blockade of the Gaza Strip despite repeated warnings from Israel to turn back.
It is about 70 nautical miles off the war-ravaged strip, inside a zone that Israel is policing to stop any boats approaching.
A live video feed from one of the boats in the flotilla showed passengers in life vests sitting on deck.
It is not clear how many of the boats had been intercepted or stopped.
Some passengers said their vessels continued to advance.
Organisers remained defiant, saying in the statement that the flotilla "will continue undeterred".
The Israeli military did not respond to a request for comment about intercepting the vessels.
The Israeli foreign ministry earlier said its navy had reached out to the flotilla to warn it was approaching an active combat zone and violating a lawful blockade, and asked them to change course.
The ministry said that it reiterated the offer to transfer any aid peacefully through safe channels to the Gaza Strip.
It is the latest sea-borne attempt to break Israel's blockade of the enclave, much of which has been turned into a wasteland by almost two years of war.
The flotilla was hoping to arrive in the Gaza Strip on Thursday morning.
This is the second time the flotilla has been approached on Wednesday.
Before dawn, the mission's organisers said two Israeli "warships" had approached fast and encircled two of the flotilla's boats.
All navigation and communication devices went down in what one organiser on board described as a "cyber attack".
A video post on the flotilla's Instagram page showed the silhouette of what appeared to be a military vessel with a gun turret near the civilian boats.
Reuters confirmed that the video was filmed from the flotilla but could not confirm the identity of the other vessel in the video or when the video was taken.
Last week the flotilla was attacked by drones, which dropped stun grenades and itching powder on the vessels, causing damage but no injuries.
Israel did not comment on that attack but has said it will use any means to prevent the boats from reaching the Gaza Strip, arguing that its naval blockade is legal as it battles Hamas militants in the coastal enclave.
Italy and Spain deployed naval ships to help with any rescue or humanitarian needs but stopped following the flotilla once it got within 278km of the strip for safety reasons.
