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"The terms of the offer are confidential, but I can ... confirm that Ocean Infinity have offered to take on the economic risk of a renewed search," the company said in an email.
"We're in a constructive dialogue with the relevant authorities and are hopeful that the offer will be accepted," it added.
Voice370, a support group for families of the 239 people on board, said Ocean Infinity "would like to be paid a reward if and only if it finds the main debris field" according to the terms of the offer made in April.
"Why hasn't Malaysia accepted this win-win offer?" Voice370 asked in a statement.
Malaysia did not immediately respond to the families' question Friday.
But Malaysia, Australia and China agree that the newly identified area is too big to justify resuming the publicly funded search, which has already cost $160 million.
Australia has coordinated the search on Malaysia's behalf because Flight 370 crashed in Australia's zone of search and rescue responsibility. It came down on March 8, 2014, after flying far off course on a journey from Kuala Lumpur, Malaysia, to Beijing.
Transport Minister Darren Chester declined to comment on the possibility of a private search.
"Malaysia, as the state of registry for the aircraft, retains overall authority for any future search and any questions regarding possible future search efforts should be directed there," his office said in a statement.
"Australia stands ready to assist the Malaysian government in any way it can," it added.