I would talk to them about it. Find out why they need you to start next week and it might make it clearer to you if there is going to be any flex about this (if they have a contractual obligation of some sort to have someone in place by Monday, they won't budge).
If there is no legal barrier to a longer transition time, then tell them you are obligated to your company to give a months notice (if you are legally obligated) or that you would prefer to give 2-4 weeks notice so that the company is not left in the lurch when you leave. Be aware that if you think giving a month's notice means you will train your successor, it is very rare that the new person would be on board anyway before you left. If you are close to a project deadline, explain that you need to be there when this project goes live next week and for at least a week afterwards. Good companies respect people who are willing to risk losing an opportunity because of their obligations to a current employer. It means you are far less likely to leave them at a bad time.
Or perhaps you can broker a compromise where you work part-time for them and part-time for your old employer if they need someone immediately and you want time to transition.
If you decide to skip the notice (and you aren't contractually required to give notice) and take the job because they won't budge, at least leave as professionally as you can. Get everything on your projects organized so someone can easily take them over (make sure everything you are working on is committed to a source control (in a separate branch if it isn't ready for prime time) and document information your replacement will need to know such as exactly where things stand and where to find certain key pieces of information) and then go take your resignation letter to your boss along with the documentation you assembled to help the transition.
Explain the lack of notice as due to the other job offer being contingent on an immediate start date and offer your contact information for them to call to if they have questions. You could even offer to work in the evenings and weekends until your replacement is hired or to train someone on your off hours once they have a replacement. Since you are not giving them the courtesy of notice, I would go the extra mile in what I could offer instead so that you don't burn bridges. In no case, leave them a mess to clean up.
Do be aware that a company that won't budge may not be the best place to work especially if there is no legal/contractual reason why they need someone next week.