According to the report, Haniyeh sent the message to Prime Minister Benjamin Netnayahu's bureau via American-born Israeli Gershon Baskin, who has in the past served as an informal mediator between the two sides, and has been credited for his role in securing the release of Israeli soldier Gilad Shalit .
Baskin told Ynet on Tuesday that he wants to "neither confirm nor deny the report", adding that he has "nothing to say."
According to the report on Walla!, Haniyeh’s adviser, Ghazi Hamad, spoke to Baskin from inside the Hamas leader's office, and with his knowledge. Baskin then proceeded to deliver the message, in writing, to Gil Shapir, the director of Netanyahu’s office.
The call apparently came after rocket attacks from Gaza and Israeli retaliations threatened to inflame the situation along the southern border.
In wake of rocket fire on Israel by radical forces in Gaza, Israel Air Force jets attacked targets in the Gaza Strip, after which Hamas withdrew the security force tasked with preventing the rockets.
The withdrawal of forces was perceived by Israel as actively facilitating further rocket fire, and thus prompted Hamas to pass on the conciliatory message and request for restraint.
On January 21, after a barrage of rocket launches at Israel, Hamas said it had deployed forces in Gaza to "preserve the truce" with Israel that ended the last major confrontation in November 2012. Though Hamas rarely fires rockets itself, Israel views it as responsible for any attack launched from Gaza.
Baskin is the director of the Israel-Palestine Center for Research and Information (IPCRI). In the first week following Shalit's kidnapping in 2006, he established an unofficial line of communication with Hamad, who served as Hamas' government deputy foreign minister.
Hamad had asked Baskin to put him in touch with Noam Shalit, a mere six days after his son was taken, and assured him that Gilad was alive and was treated according to Islam's laws regarding prisoners of war, in other words, he had been given shelter, food and medical care.
Hamad and Baskin maintained their communication throughout Shalit's five years in captivity, mostly via email and phone calls. According to recent reports, the two are still in contact.
Hamas on Tuesday denied that its prime minister in Gaza, Ismail Haniyeh, had sent Prime Minister Benjamin Netanyahu a message via an Israeli intermediary, claiming that the group was not interested in an escalation of hostilities in Gaza.
Gaza Deputy Foreign Minister Ghazi Hamad told Turkish news agency Anadolu that the "claims are completely fabricated."Elior Levy and AFP contributed to this report