Hi Sushi....First of all , do you have any children ? If so that's a different scenario. If not then I was told , and found out the hard way that going no contact is the best way to move forward. It seems he still wants to be part of your life. It will probably hurt your new relationship. In my experience there's no way you and your ex can remain " friends ". It'll always be more than that. Only communicate with him if you absolutely have to. As harsh as it sounds , no contact, is the way you have to go so you can get on with your life.
I hope things work out well for you...