Moyes was sacked as United manager on Tuesday, less than 12 months into a six-year contract, with the reigning Premier League champions languishing in seventh place in the table and unable to qualify for next year's UEFA Champions League.
Club legend Ryan Giggs has since been appointed as interim manager at Old Trafford, taking charge of his first game against Norwich City on Saturday.
In a brief statement via the League Managers' Association earlier this week, Moyes had thanked United fans for their support during his time in charge.
And according to United We Stand fanzine editor Andy Mitten, he also received a text from Moyes late on Friday evening expressing gratitude to the club's followers.
In a message posted on Twitter on Saturday, Mitten wrote: "Got a text from David Moyes last night: 'Andy. Would you please let it be known how much I appreciated the support I got from the real United fans. They were incredible. I am sorry I couldn't give them the results they are all used to. Thanks. D Moyes'."
While Moyes himself has not spoken publicly other than via the LMA, there has been widespread praise for the dignified manner in which the Scot has dealt with his Old Trafford exit.
Several fellow Premier League managers also had their say when asked about his departure ahead of the weekend's action, with Arsenal boss Arsene Wenger conceding that he was "sad" that Moyes had not been given more time.
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