Israel has so far allowed two of six $15 million payments, intended for the poor, to reach Gaza. The government, however, halted the third amount after officials said a terrorist shot an Israel Defense Forces soldier dead in the Palestinian territory.
"Hamas has renewed its call for mediators to exert pressure on the 'occupation' to implement the understandings soon, due to the tense situation in the Gaza Strip," a Hamas member said.
"A boiling volcano will erupt in front of the enemy and its soldiers, and escalation is something [Prime Minister Benjamin] Netanyahu does not want before the elections."
Qatari representative Mohammed Al-Amadi is expected to meet soon with Israeli and Hamas officials to talk about the donated funds.
Hamas has blamed several attacks on Israelis on "lone-wolf" Palestinians frustrated with economic conditions in Gaza. Its leaders have said anger among citizens about their humanitarian conditions could lead to war.
The Hamas threat comes less than a month before Israel's second national election on Sept. 17. Netanyahu failed to form a coalition government in May after his Likud Party narrowly defended a challenge from the centrist Blue and White Party.
"This is what extortion looks like," the party said. "To anyone who still doesn't get it: our deterrence hasn't eroded, it's collapsed. In a Blue and White cabinet, we will set the agenda for Hamas, and won't be its supplier of dollars."