Hi, SitaClare, Welcome to the forum!
Are you planning to walk the Camino de Madrid from Madrid? Or are you just saying you will be arriving in Madrid and you want to start walking on the Camino de Levante? I agree with JLWV that it would be a shame to start in Avila and miss the mountain days before it. If you have enough time, you could take a bus or train from Madrid to Toledo and start walking on the Levante there. If you don't know Toledo that would be a great opportunity to see it, it is a beautiful town chock full of things to see, but also equally full of tourists. That's why walking from there is a great idea, because it would give you a night to enjoy the peace and beauty of Toledo after dark.
If you want to walk the Camino de Madrid, you could walk the first few days to Segovia, and then take a bus from there to Avila to connect with the Levante, but I think the first suggestion is the nicer one. Buen camino, Laurie