Think what you want to achieve. First, you don't want to end up without a job. Second, you want the job with the new company. Third, you don't want to get sued by the old company.
The rule is that you give notice when you have a signed contract with the new company. If you talk to your old company earlier (like "I have an offer from another company, would it be possible to leave with only two weeks notice"), there's the risk that the old company says "Don't worry about notice, you're fired" and then the new company says "we changed our mind and hired someone else". So you can't do that.
You can't sign a contract to start in two weeks if you need to give 30 days notice, because that means you can get sued by the old company. That's completely out of the question.
The only way you can start with the new company is if they offer you a job with a start date 30 days from now. In addition, you can agree to start earlier if you can reduce the notice period, so with this contract in hand you can give notice and ask for a shorter notice period, or the new company can call the old company and offer them cash to let you go earlier.
If they insist on a start date in the contract that is earlier than your notice period ends, then they are entirely unreasonable, and there is no way you can accept their job offer. One, because you will be in trouble, two because you don't want to start with a company that proves itself to be unreasonable before you even start.