"Some Saudi intelligence officials officially contacted some Hamas leaders outside the Gaza Strip regarding recent developments in relations between Hamas and Iran," the Palestinian Al-Ray media agency, which was previously run by the Gaza Strip's Hamas-led government, quoted a source as saying.
The source added that Saudi Arabia had expressed anger over Hamas' policies toward Iran, calling on the Palestinian faction to suspend all communication with the Islamic republic.
He said Saudi Arabia had promised in return to ratchet up pressure to lift Israel's blockade – in place since 2007 – on the Gaza Strip.
"Hamas leaders received the Saudi request, but have yet to reply; they've promised to consider it," the source told the news agency.
A Hamas delegation visited Tehran last week with a view to improving relations with Iran.
Relations between Hamas and Iran had deteriorated markedly due to Hamas' position on the ongoing civil conflict in Syria.
Tension has marred ties between Saudi Arabia and Iran against a backdrop of political and sectarian considerations. Iranian support for the government in Syria has only served to widen the gap between Riyadh and Tehran.
Hamas used to have strong ties with Iran before the eruption of what came to be called the "Arab Spring" uprisings.
When the uprising erupted in 2011 against the regime of Syrian President Bashar al-Assad, Hamas refused to express support for the latter, which is backed by Iran.
Recent months, however, have seen Hamas and Iran taking tentative steps towards reconciliation.
On Sunday, Hamas' armed wing, the Ezzedin al-Qassam Brigades, thanked Iran for providing it with "financial and military" assistance used to defend the Gaza Strip during Israel's recent 51-day onslaught on the embattled enclave.
Speaking at a Gaza City rally on Sunday, Brigades spokesman Abu Obeida thanked Iran for its "support of the Palestinian resistance" by providing rockets that had been used against Israeli targets during the self-proclaimed Jewish state's deadly summer offensive.
Abu Obeida also expressed thanks to Qatar and Turkey for "supporting [Gaza's] people and the Palestinian cause."