The Red Devils announced last week it had agreed a deal in principle to sign the France centre-back, who had a year left on his contract with LaLiga giant Madrid.
United is reported to be paying a fee of £42.7 million ($80 million) to secure the transfer, with the player set to travel to England this week, where he will undergo a period of isolation before completing a medical.
Varane visited Madrid's Valdebebas training ground on Saturday (AEST) for the final time before departing the club where he won four UEFA Champions Leagues and three LaLiga titles among a host of trophies after his arrival from Lens in 2011.
"These past few days have been charged with many emotions, feelings that I now want to share with all of you," he wrote on Instagram.
"After 10 incredible and wonderful years at Real Madrid, a club I will always carry in my heart, the day has come to say goodbye.
"Since I arrived in 2011, together we exceeded all expectations and achieved things I could never have dreamed of.
"I would like to thank all the coaches and all the people who work or have worked for the club for everything they have done for me.
"Many thanks also to all the Madridistas who always gave me a lot of affection and with their great demands pushed me to give my best and to fight for every success.
"I have had the honour of sharing a dressing room with the best players in the world. Countless victories I will never forget, especially 'La Decima'. I realise it's been a great privilege to have been able to experience such special moments.
"Finally, I want to thank all Spaniards and especially the city of Madrid, where my two children were born. This country will always be special to me.
"It's been an incredible journey in every way. I leave with the feeling of having given everything and I will not change a single thing in this story of ours.
"A new chapter begins..."