The American company, Ocean Infinity as well as Dutch outfit Fugro, which had been involved in the original search, and an unidentified Malaysian company had put forward proposals to relaunch the hunt.
Ocean Infinity was reported to be favoured after making a "no find, no fee" offer to search for the Malaysia Airlines plane.
The jet disappeared with 239 people on board in March 2014 en route from Kuala Lumpur to Beijing after diverting from its flight path.
No sign of the plane was found in a 120,000 square kilometre (46,000 square mile) zone selected by satellite analysis of the jet's likely trajectory.
The Australian-led hunt -- the largest in history -- was suspended in January, sparking criticism from families of those on board and some experts, who said it was called off too soon.
On Thursday, family members of passengers were sent a message by the MH370 Response Team in Malaysia saying they were in talks with Ocean Infinity on the terms of an agreement.
"The MH370 Response Team has received several proposals from interested parties to search for MH370," said the emailed message, a copy of which was seen by AFP.
"These offers have been thoroughly assessed by the team and the governments of Australia and China... The government of Malaysia has given the permission for the response team to proceed negotiating the terms and conditions with Ocean Infinity."
"The ministry is still negotiating to finalise the terms... We favour Ocean Infinity," he told AFP.
Danica Weeks, an Australian whose husband Paul was on the flight, told AFP she was "ecstatic that the Malaysian government is doing what they need to do to continue to find MH370".
Ocean Infinity, which said in a statement that "good progress has been made" in negotiating the contract, claims it has the world's largest and most advanced commercial fleet of underwater vehicles for conducting searches.
"We remain optimistic that we will be able to try and help provide some answers to those who have been affected by this tragedy," a spokesman said in a statement to AFP.
Only three confirmed fragments of MH370 have been found, all of them on western Indian Ocean shores, including a two-metre wing part known as a flaperon.
(This story has not been edited by NDTV staff and is auto-generated from a syndicated feed.)