SYDNEY—The best leads in an underwater search for Malaysia Airlines Flight 370 will be exhausted in about a week, Australia's Prime Minister Tony Abbott said Wednesday, as searchers battled to overcome technical issues that have hampered early efforts to scan the Indian Ocean's seabed.

Mr. Abbott said authorities would need to rethink their approach if a remote-controlled vehicle fails to locate wreckage from Flight 370 in a narrow area of ocean where searchers earlier this month picked up the strongest electronic signal out...