Some background, I'm an intern at a big company in the US. When I got my internship over a year ago, I did so with the understanding I'd get offered a job once I graduated. In the last few months, it's been made clear that getting a full time position is slim, and the only position available has dismal pay and isn't what I want to do. I started job searching and got offered a good job with a salary I would want. However, this position starts once I graduate, in January (2 months from now). My boss has been very helpful in trying to get me in to one of these full time positions, and has been asking about these positions often. How do I handle these questions? Do I tell them my plans to leave the company in January? I don't want my boss to end up looking foolish from trying to get me one of these positions, only for me to turn them down.
Your boss could potentially find you a better position, so don't turn down a potential job because you believe it'll be worse. Potentially, the pay may be lower but have better benefits. Let your boss refer you to a potential position, and politely turn it down if it doesn't meet your expectations.
As far as your notice period goes, understand what you're required to give and give just that. If your contract requires a notice period, give your notice at the beginning of the appropriate time period. If it doesn't, then you can give the standard 2 weeks notice. If you don't believe your boss will respect your notice, you're free to quit without notice but may suffer a bad referral as a result. Some contracts simply expire without a renewal clause, in which case you don't need to give any notice.
The best advice to keep your current position is to carry on as if everything is normal. Sure the boss may continue to try to land you a full time position at your present company and you can play along with that, however when the time of you internship comes to an end you can leave at will.