No. Competition exists when the rules are the same for all parties.
Illegal immigrants don't have 1/3 of their pay taken up-front in payroll taxes. A mexican working for 10 dollars an hour takes home 10 dollars per hour. An American citizen has to "make" 15 dollars per hour to take home 10 dollars per hour. In the higher-tax states the disparity is greater.
American workers are further handicapped in the "competition" when the illegal immigrants have free access to government services that are not freely available to the employed American -- services that the American must pay for through taxes to benefit the invaders, then pay for again when they need to use the services.
American workers don't get free college, free medical, interest-free SBA loans, free housing, along with tax-free living, that the immigrants enjoy.