Yes, it is common for companies to contact previous employers during the reference check process. However, they are typically only checking employment dates and job titles to verify your work experience. It is illegal for a company to give false or defamatory information about an employee, so your previous employer should not speak negatively about you. If they do, you may have legal recourse. As for your current situation, you can try to negotiate a shorter notice period with your employer or explain your situation to your new employer. It is important to be upfront and honest in all job-related matters.