The 22-year-old has impressed during a two-week trial to earn a contract with the League One club.
He told Magpie Player: "It was a big chance for me to come here and luckily the manager's liked me and given me a chance.
"It went well for me and now I've got a contract at a fantastic club like Notts County.
"(Manager) Chris (Kiwomya) has been very good to me and has given me a lot of advice and a lot of belief in myself, which sometimes you can lack.
"I feel very confident at the moment and I know the manager likes me so hopefully he'll give me some minutes on the pitch and I can repay him with goals.
"The most important thing is that we have a good season and win so I'm willing to do anything it takes to do that.
"I want to be in the best shape I can, score goals and hopefully take the team up to the Championship."
Coombes, formerly known as Adam Phillip, has represented England at Under-16 and Under-17 levels and has previous experience of League One football with Yeovil.
He will go straight into the Magpies squad for their opening game of the season away at Sheffield United on Friday night.