The out-of-contract duo were both offered new terms by the Daggers at the end of last season, but Bingham has yet to respond while Doe has confirmed he will not be signing.
Boss Wayne Burnett told the club’s official website: “I spoke to Billy at the beginning of May and we offered him a very good deal, we were very keen for him to stay here. We agreed he’d go away and think about it as he wasn’t entirely sure what he wanted to do.
“We are keen to keep him, I got a call from his agent about 10 days ago saying that he was still unsure and he was going to speak to some clubs and I haven’t heard anything since.
“I’ve tried to call him a couple of times but to no avail. For us, we’ve given him a very good deal, we’ll have to wait and see.
“Scott Doe has declined our terms, Billy hasn’t turned them down but we just don’t know whether he’s accepted them.
“Scott has decided it’s time for him to move on. We wanted to keep him but he wanted to try something new, which we are all disappointed about. We wish him well, he’s been fantastic for this football club.”