Can you suggest to me the best way to deal with this situation: Career Change?

I have found another job at a reputable company. They have checked my references and it all looks good. The only thing left to do is contact my current boss and do my criminal background check. I don't want my current employer to know that I am looking to leave the company until I have a firm offer from the new job. What can I do about this? Any suggestions on how to go about this? Thanks in advance for your answers.Can you suggest to me the best way to deal with this situation: Career Change?
I am very surprised that the new job wants to contact your present job. That is very unusual. Just so you know, the new job can only ask your present job how long you have been there. They can't ask ';So how is this person doing?'; Meaning they are not allowed to ask ';dirty'; questions about you. If possible, explain to the new job you would be happy to prove that you have worked there that long. (provide pay stubs, or tell your present employer you need a letter on company letter head for an apartment or something.) If the the new job hasn't made a firm promise to you about your future job, it's not a good thing. Normally the new job would hire you, then call the old job to get info. If you lied to get the job, they would fire you.