The left-back is poised to move to Villa Park for a fee of around £25million, the PA news agency understands.
Digne has fallen out with Everton boss Rafael Benitez, who confirmed last week the 28-year-old no longer wanted to play for the Toffees.
Digne, signed by Everton from Barcelona in August 2018, said in a post on Instagram on Wednesday evening that he has “loved to be a part of the Everton family”.
And the France international added: “Only one year ago I signed a new contract with the ambition of staying in this club for a long time, giving everything for my club, for the development and project, that I believed in – and for the passionate fans.
“My dream was helping the club back where it belongs. Wearing the captain’s armband in some matches always made me proud.
“Everything must come to an end. I just did not expect it to end this way. What has happened and some things that was said about me in the last month has made me very sad. But I will not enter a war on words with anyone.
Everything must come to an end. I just did not expect it to end this way. What has happened and some things that was said about me in the last month has made me very sad
“The club don’t deserve that, the fans don’t deserve that – and to be honest, I don’t feel that I deserve that. There are so many good and decent people in and around Everton who want the best for this club and not only for themself – and to them I can only say from my heart: I wish you the best!
“Thank you, to all you wonderful, proud and passionate true Evertonians. It has been an honour and a true pleasure to play for you, the fans, the people – because a club do not belong to a player or a manager, but to the fans. I will always carry you with me in my heart wherever I go.
“Sometimes it only takes one person from outside to destroy a beautiful love affair. All the best to @everton and Evertonians.”
Villa have already signed Philippe Coutinho on loan from Barcelona as Gerrard makes his mark on the squad after replacing Dean Smith in November.
Gerrard had been open about wanting to sign a full-back to compete with Matt Targett.
Digne had been linked with Newcastle and Chelsea but Villa have won the race for the France international.
The defender has not played since the Merseyside derby defeat on December 1 after a fall-out with the manager over tactics and his role in the team.
He returned to the squad as an unused substitute a month later for the defeat at home to Brighton but last Friday the Toffees boss revealed the player had asked to leave.
Benitez was critical of Digne’s attitude, saying: “We are paid big money to do our job and if someone is happy or not happy it is (still) up to you to perform in your job.
“You have to put your interests behind the interests of the club. If you are not thinking about that the priority is you and not the team.”
Villa winger Anwar El Ghazi could yet move to Goodison Park but the deals are not linked.