Thanks. (PS4)
Have you got that Castle quest active? If it's active he might get stuck on asking you to go sort it, and not give option to be your companion...so fixing this would mean completing that quest.
He tends to bug after the quest for a lot of people. I know since taking the Castle and having him wander back to Sanctuary, all he does is spout radiant quests at me and I never see conversation options with him anymore.
My fiance finished it and Preston works just fine, so it's just buggy. There are a bloody ton of companion bugs that need fixing >.>
To solve your problem, your options are "Reload a previous save before he was bugged" and "Well, you could start a new game"
EDIT: Or, like Tawnee said, if he's just going on and on about taking back the Castle and meeting you there, go ahead and go through with it. But he still might bug afterwards
Check to see if you have any settlement quests open. If you have too many, you may need to close one or two out. I can't guarantee this will solve the problem, but I noticed his dialogue changed after I completed two settlement missions. I don't take companions, but he did offer.