The weekend meeting is the first publicly reported since the start of the Iran war between the Palestinian militant group and the board, a new international body personally headed by Trump, which has been tasked with overseeing post-war Gaza.
Following the meeting, Israel announced on Sunday that it would soon reopen the sole crossing for pedestrians between the Gaza Strip and Egypt, shut since the Iran bombing campaign began.
One of the sources said he believed the Israeli announcement was a direct result of the meeting between Hamas and the board.
Prior to the war in Iran, Trump's plan for the Gaza Strip was his flagship initiative for the Middle East.
The sources said the Hamas representatives warned the board that the militant group could back away from its previous promises under the enclave's ceasefire if Israel maintains new restrictions on the Gaza Strip imposed during the Iran war.
Israel shut the strip's borders after the war's launch on February 28, saying crossings could not be operated safely.
It later resumed the limited flow of goods and aid but kept shut the sole crossing for pedestrians into Egypt, at Rafah on the Gaza Strip's southern edge.
On Sunday it announced it would reopen the crossing later this week following a "security assessment".
One of the sources said Trump's board was represented at the talks with Hamas by Aryeh Lightstone, a US aide to Trump's special envoy Steve Witkoff.
The other two sources said meetings that included Lightstone were on the agenda although they were unable to confirm whether Lightstone had attended yet.
Further meetings were expected this week.
A US official said that Lightstone had attended Gaza Strip-related meetings in Cairo in the past several days, without confirming whether he met with the Hamas delegation.
US negotiators were continuing to meet with regional partners to fulfil Trump's 20-point plan for the enclave, the US official added.
A Palestinian official with knowledge of the Cairo talks said that Hamas believed Israel was exploiting the war on Iran to slip away from its obligations under Trump's plan. Israel rejects this.
Trump's plan has hinged in part on whether Hamas militants would lay down their arms in exchange for amnesty, a step intended to pave the way for reconstruction and further Israeli military withdrawals.
None of the sources said whether disarmament would be discussed in Cairo talks this week.
