Tony Blair is understood to be acting as a go-between in secret talks aimed at lifting the Israeli siege of Gaza.
Since resigning as the Quartet's Middle East envoy in May, the former prime minister has twice held talks with Hamas leader Khaled Meshaal, according to Arabic media reports, while Israel's Haaretz newspaper said Mr Blair had earlier discussed his plans with Binyamin Netanyahu, the Israeli prime minister.
As the Quartet envoy Mr Blair would have been prevented from holding talks with Hamas, which has run the Gaza Strip since 2007.
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The meetings have taken place in Doha, Qatar and on the agenda has been a temporary ceasefire deal with Israel that would lift the siege of Gaza, according to reports.
According to Al-Hayat, the Arabic-language daily, under the deal Isreal would lift its blockade of Gaza and allow the establishment of a naval corridor between Cyprus and Gaza.
In return, Hamas would agree to a long-term ceasefire lasting as much as 10 years.
Comments this week by Yasin Aktay, a senior adviser to Turkish prime minister Ahmet Davutoglu, appear to confirm the talks between Hamas and Mr Blair.
Mr Aktay said that, on a visit to Turkey, Mr Meshaal had discussed details of the truce as mediated by Mr Blair.
Turkey, which has a strained relationship with Israel following Israel's killing of nine Turkish citizens in a 2010 raid on a flotilla trying to break the blockade of Gaza, has been involved in the Doha talks.
But as speculation in Arabic media grew, Israel's Mr Netanyahu denied that he had authorised secret talks and said Israel was not in contact with Hamas either directly or through third parties.
Rumours of secret talks have surface before to little effect.
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