Meshaal, who is based in Damascus, the Syrian capital, spoke as he made his first visit to the Egyptian capital in months.
The recent raids in the West Bank, which prompted deadly violence in the city of Qalqilya, have fuelled concerns of a wider conflagration between armed units of Fatah and Hamas.
Meshaal said Hamas and Egyptian officials had discussed how to resolve the crisis, examining options such as freeing Hamas political detainees and ending the security crackdown in the West Bank.
Meshaal on Tuesday welcomed the criticism of Israel's settlement expansion by the administration of Barack Obama, the US president, but stressed that what was needed were "actions on the ground, not just the tone."
He said that Suleiman had told him there is an "encouraging vibe" from Washington, but the Hamas leader stressed that a future Palestinian state must be based on the 1967 borders between the Palestinian territories and Israel.
The US must also respect Palestinian demands regarding the future of east Jerusalem and the right of return for Palestinian refugees who fled their homes during the previous Arab-Israeli wars.
Meshaal called on the international community to place serious pressure on Israel to freeze its settlement activity, which is illegal under international law.
The talks between Egypt and Hamas come two days after Egyptian officials met Ahmed Qurei, who leads Fatah negotiators in the Cairo-sponsored reconciliation talks.
Previous rounds of talks between Fatah and Hamas have failed to solve major differences between the two factions.
While Fatah has moved against Hamas supporters in the West Bank, it has accused Hamas security forces in Gaza of detaining dozens of its supporters.
Egyptian mediators are trying to convince the groups to form a unity government, setting a July 7 deadline to reach a deal.