I am currently on my notice period because I accepted offer from another company. I don't want them to call my current company and ask for references (i have no problems if they call to my previous companies).
Could asking not to call to the current one have negative impact for the hiring decision of new company?
I'm not sure where you are based, but from my experience there are 2 different things here:
The employer is seeking a reference, this does still happen (in my experience in smaller companies), but usually this happens BEFORE an offer is made. In this you don't really have an option, the employer will expect good feedback from your current employer, if you are concerned about what they say, you need to get onto this NOW, and pre-empt it, possibly suggesting a different party who can vouch for you. But beware, the fact your current employer won't will be a red flag to them (unless the outgoing employer is someone they know to be suspect).
What is more likely given the position you describe, is employment verification, where they will simply be checking you have the title/role you say and dates. There is usually a clause in your contract joining that they have the right to withdraw if an anomaly comes up in verification.
In my experience (having worked in enhanced verification areas like police and banks), the second option is probably what is happening, and the likelihood is that you don't really need to avoid it, and if it is the other situation I don't think you really can avoid it.
The general rule is that employers don't ask references from a current employer. The reason for that is that if you're a bad employee, they'll lie about you to make you someone else's problem, and if you are a good one, they'll lie to keep you or out of revenge. I have personally witnessed this more than once.
It is also policy at some companies that employees are not allowed to give out any information about current employees. HR departments know this so they usually skip the current employer, except for details of employment such as start date, end date, and title.