World football’s governing body Fifa have granted the clubs until 15 April to settle the row.
Cardiff refused to make the first scheduled payment for the Argentine.
Fifa had originally ordered evidence to be submitted by Wednesday to adjudicate in the dispute.
Cardiff say Nantes have yet to respond to their request, sent last week, to meet over the issue.
A club statement said: “Cardiff City has requested and been granted an extension to the deadline to respond to Fifa on this matter.
“Cardiff City recently wrote to FC Nantes proposing a meeting to discuss issues surrounding the Emiliano Sala tragedy and planned transfer, in line with Fifa’s request for our two clubs to come to a resolution directly.
The Bluebirds have contended the agreement struck with Nantes to buy Sala was not legally binding.
They say conditions, which they claim were insisted on by the French club for completion of the deal, were not fulfilled.
Nantes have insisted they completed all the necessary paperwork and had been fully compliant with Fifa’s rules.
Fifa president Gianni Infantino has said he hoped the two clubs would be able to reach an agreement between themselves.
Now Cardiff have revealed they have moved to offer discussions to Nantes.
Sala, 28, was announced as Cardiff’s record signing on 19 January.
He died in a plane crash late on 21 January near Guernsey, when he was a passenger on a flight from Nantes to the Welsh capital.
The aircraft was piloted by David Ibbotson, who is still missing. BBC Wales understands Ibbotson was not qualified to fly at night.
He is thought to have been colour blind and his licence restricted him to flying only in daytime hours.