Question about applying: going to new job for 3 months when I apply to consulting

TL;DR: My current job to expire soon, and I received a job offer elsewhere (job starts mid-July). However, I want to apply to consulting firm. Can I apply to consulting firm w/o letting them know that I am working my new job?

Hi folks,

Thought I might ask this question here. I worked at a temp job for the last 3 years (I'll call Company A), and my job is set to expire this August/September. Thus, I applied for a couple of jobs and received a job offer from another company (I'll call Company B) with a start date of mid-July.

My desire is to apply and work at a MBB consulting firm (MBB). The recruiter for MBB said that the application process takes 2-3 months, and that the MBB generally frowns upon candidates who did not stay at their "current" job for at least a year.

I absolutely want to apply and work for the MBB. However, I also do not want to renege on my current job offer from Company B in case my consulting application does not work out and my current job's term at Company A expires.

Thus: If I were to apply right now to MBB before I quit my job at Company A, my resume and application will still show that I work at Company A. Can I work in Company B from mid-July onwards throughout my entire application process of 2-3 months without MBB knowing that I switched jobs? Thus, will MBB still think I am working in Company A in my current position? And if I am fortunate to receive an offer from the MBB, will they investigate/verify again to see if I am still working at Company A (to which by then I will be gone)?

My supervisor in Company B is OK with me leaving after working for a couple months. But I want to know if the MBB firm can see me for only my position at Company A and not ding me for staying at Company B for only a couple months.

Thank you all.