The search for Argentine Sala, 28, and 59-year-old pilot David Ibbotson was called off on Thursday.
Their plane disappeared from radar as they flew over the English Channel on their way to Cardiff on Monday night.
McKay has revealed text messages with Sala discussing flight arrangements.
One of McKay’s sons, Mark, was the agent acting for Nantes in the £15m transfer that made Sala Cardiff’s record signing earlier in January.
McKay told the BBC he had arranged the flight through David Henderson – an experienced pilot who had flown him and many of his players “all over Europe on countless occasions”.
It is not known how the job of flying Sala ended up being passed to Ibbotson.
The Civil Aviation Authority is to examine whether Ibbotson was operating with a private or commercial licence.
Henderson has not so far commented on what happened.
In the text exchange released to the BBC by McKay, his other son, Jack – who is a Cardiff player – offers to arrange the private flight for free so Sala could return to Nantes to pick up his belongings and say goodbye to former team-mates at the Ligue 1 club.
McKay said both Mark and Jack had a conversation with Sala about the difficulties of flying to the French city by commercial airline.
He also says a Cardiff player liaison official knew of the arrangements that were made. The BBC has approached the Premier League club for comment.
“With regards to the booking of the flight we contacted Mr David Henderson, who has flown us and many of our players all over Europe on countless occasions,” McKay said.
In a 2015 interview with the BBC, Mr Henderson spoke about his life as an experienced light aircraft pilot.
McKay added: “We had no involvement in selecting a plane or a pilot and we also wish to make clear again we do not own the plane that Emiliano flew on.”
The Times newspaper has apologised to McKay for incorrectly stating that he owned the plane.
The agent also said he has met Sala’s friends and family to “give them an understanding of how Emiliano came to be on that plane”.
“The tragic events that have unfolded have shocked us all,” he added.
The Air Accidents Investigation Branch has started an investigation which will look at “all operational aspects,” including licensing and flight plans.
Similarly the Civil Aviation Authority is to examine whether the pilot was operating with a private or commercial licence.
Cardiff have started an internal inquiry to try to establish the events leading up to the flight, but a senior source has denied reports the club is considering a multi-million pound negligence lawsuit over the incident.
The source claimed senior club officials only became aware a private flight had been arranged after they were alerted by the club’s player liaison officer that Sala had failed to arrive at Cardiff airport as planned on Monday.
They also made the point that the club does not have a private jet for players to use and therefore they could not be expected to have arranged his travel to and from Nantes.
Argentina’s president has joined calls from Sala’s family and many fellow footballers for the search to resume.
Cardiff’s players will wear yellow daffodils for Tuesday’s Premier League fixture at Arsenal to honour Sala.
Full text message exchange
These messages between Jack McKay and Sala were translated from French.
Friday, 18 January
7:43pm – Jack McKay:“My dad has told me that you are going home tomorrow. He could organise a plane to take you direct to Nantes and to come back on Monday, at a time that suits you, so you can get to training on Tuesday.”
7:51pm–Emiliano Sala: “Ah that is great. I was just in the middle of checking if there are some flights to get to Nantes tomorrow.”
7:56pm–McKay: “He said he could organise a plane that would go direct to Nantes.”
7:56pm– Sala: “How much will it cost?”
7:56pm–McKay: “Nothing. He said if you help me to score goals it’s nothing.”
7:59pm–Sala: “Hahaha with pleasure.”
8:00pm– Sala: “We are going to score lots of goals.”
8:01pm – Sala: “I want to leave tomorrow for Nantes at around 11am and come back on Monday night around 9pm to Cardiff if that is possible.”
8:05pm –McKay:“Good. I’ll send a message when that’s sorted.”
Sunday, 20 January
5:00pm – McKay: “Hi there is it possible you could come back at seven in the evening on Monday night? Just because the pilot has to get home in the north after he gets to Cardiff.”
5:01pm – Sala:“Hi, Half past seven would be possible.”
5:03pm – McKay:“Yes that’s good.”
5:05pm –Sala: “[PICTURE OF LUGGAGE] Can you ask if I can bring this on the plane?”
5:06pm –McKay: “Good yeah.”
5:07pm – Sala:“But is that going to be OK for the plane?”
McKay: “Yes there is space on the plane for your luggage.”
5:12pm – Sala: “OK.”
Monday, 21 January
4:16pm – McKay: “I’m going to call in a moment.”
4.23pm –McKay: “He said that it is the same company.”
4.27pm – Sala:“OK thanks.”