Sala's body was recovered from the English Channel and formally identified on Thursday evening, nearly three weeks after his plane crashed en route to Cardiff - whom he joined for a club-record €17 million.
The 28-year-old was traveling to Cardiff in a plane piloted by David Ibbotson, which went missing over the English Channel last month.
In a statement, they said: "We would like to thank you for all your signs of affection and support in what is the most painful time of our lives".
On Thursday, the agonizing search for Sala ended when Dorset police confirmed that the body recovered from the wreckage of a plane the day before was the soccer player.
Some of the club's players reacted on Twitter.
They added that their thoughts are with the family of pilot David Ibbotson.
The Piper Malibu plane, registration N264DB, was found 67 metres underwater 21 miles off the coast of Guernsey and weather conditions meant that investigations were delayed.
Nantes have allegedly threatened to take legal action if Cardiff do not pay them within 10 days, although reports from Cardiff suggest the club will honour their agreement with the Ligue 1 club.
According to the Evening Standard, Sala put Nala in a kennel in Nantes before boarding the plane. The search was initially called off by the Guernsey harbor master on January 24, but resumed amid pleas from Sala's family and the support of several soccer stars, like Barcelona's Lionel Messi, Paris Saint-Germain's Kylian Mbappe and Manchester City's Ilkay Gundogan.
The AAIB said that extensive video footage captured of the sunken wreckage is expected to "provide valuable evidence for our safety investigation".
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