Wenger's 22-year reign at Arsenal will conclude at the end of the season, after three Premier League titles and seven FA Cup crowns.
Chile international Sanchez left Arsenal in January, having been lured to the Gunners from LaLiga giant Barcelona by Wenger in 2014.
Sanchez paid tribute to his former manager after scoring in United's 2-1 win over Tottenham Hotspur in Sunday's (AEST) FA Cup semi-final.
"I have already written to him, not on social media but I wrote to him personally, I sent him my congratulations and he thanked me," Sanchez said.
"It's sad because he is a football legend; a gentleman who shows respect and professionalism, and I only have words of gratitude for him."
Sanchez produced a man-of-the-match display to sink Spurs at Wembley, his header cancelling out Dele Alli's opener before Ander Herrera put United ahead in the second half.
It was Sanchez's third goal for United since swapping London for Manchester, and a result that would have no doubt made his former Arsenal fans happy.