Two of the countries, Hungary and Vietnam, said they have accepted.
Australian Prime Minister Anthony Albanese said he had received correspondence from the White House overnight and would consider the request.
"We'll consider all of these approaches respectfully and through our proper processes," Mr Albanese told ABC radio on Monday.
A $1 billion contribution secures permanent membership on the Trump-led board instead of a three-year appointment, which has no contribution requirement, according to a US official who spoke on condition of anonymity about the charter, which hasn't been made public.
The official said the money raised would go to rebuilding Gaza.
Hungarian Prime Minister Viktor Orbán has accepted an invitation to join the board, Foreign Minister Péter Szijjártó told state radio Sunday. Orbán is one of Trump's most ardent supporters in Europe.
Vietnam's Communist Party chief, To Lam, also has accepted, a foreign ministry statement said.
India has received an invitation, a senior government official with knowledge of the matter said, speaking on condition of anonymity as the information hadn't been made public by authorities.
Jordan, Greece, Cyprus and Pakistan said Sunday they had received invitations. Canada, Turkey, Egypt, Paraguay, Argentina and Albania have already said they were invited. It was not clear how many have been invited in all.
The US is expected to announce its official list of members in the coming days, likely during the World Economic Forum meeting in Davos, Switzerland.
Those on the board will oversee next steps in Gaza as the ceasefire that took effect on October 10 moves into its challenging second phase. It includes a new Palestinian committee in Gaza, the deployment of an international security force, disarmament of Hamas and reconstruction of the war-battered territory.
In letters sent Friday to world leaders inviting them to be "founding members", Trump said the Board of Peace would "embark on a bold new approach to resolving global conflict".
That could become a potential rival to the UN Security Council, the most powerful body of the global entity created in the wake of World War II.
The 15-seat council has been blocked by US vetoes from taking action to end the war in Gaza, while the UN's clout has been diminished by major funding cuts by the Trump administration and other donors.
Trump's invitation letters for the Board of Peace noted that the Security Council had endorsed the US 20-point Gaza ceasefire plan, which includes the board's creation. The letters were posted on social media by some invitees.
The White House last week also announced an executive committee of leaders who will carry out the Board of Peace's vision, but Israel on Saturday objected that the committee "was not coordinated with Israel and is contrary to its policy," without details. The statement by Prime Minister Benjamin Netanyahu's office was rare criticism of its close ally in Washington.
The executive committee's members include US Secretary of State Marco Rubio, Trump envoy Steve Witkoff, Trump's son-in-law Jared Kushner, former British Prime Minister Tony Blair, World Bank President Ajay Banga and Trump's deputy national security adviser Robert Gabriel, along with an Israeli business owner, billionaire Yakir Gabay.
with AAP
