I disagree Riverwalk, the OP is a single Mum and I should hope that her and her children come as a package.
What's the point of becoming serious with someone if they don't get on with your kids and visa versa.
At the moment, she is just meeting this guy as a friend, there is no romance going on, they are just getting to know each other and quite rightly each other's children.
It may go no further than friendship, so there's no problem in them meeting with their respective children, as friends. IMO