An Israeli diplomatic source said Blair was informed by Ehud Barak, Israel's defence minister, "that there was a threat that could materialise as he entered the Gaza Strip".
But Sami Abu Zuhri, Hamas' spokesman, said the group's security forces had taken measures to ensure Blair's safety.
"The story of security threats was only an Israeli invention to prevent the visit," Abu Zuhri said.
"Blair should have not accepted these silly allegations."
Ismail Haniya, the de facto Hamas leader in Gaza, also held the Israeli government responsible for the cancellation of Blair's visit.
'Keen to go'
Blair told the Palestinian news agency Ma'an he had been keen to go ahead with the trip despite the warnings, but cancelled it after learning the threats were "more specific and more credible".
He still planned to visit Gaza at a later time, Blair told Ma'an.
"I intend to go as soon as I can. It would have been important to go and see for myself first-hand what's happening in Gaza and I will continue to press for help for the people there," Blair said.
Blair had not planned to meet any Hamas leaders, which was a decision made in line with a US-led boycott of the group over its refusal to recognise Israel and renounce violence after it won elections in 2006, Palestinian and Western officials said.
The Quartet of international mediators - the US, EU, Russia and the UN - appointed Blair to the post of Middle East envoy a year ago to bolster chances for an Israeli-Palestinian peace deal this year.
Blair last visited the Gaza Strip as the UK prime minister in 1998.